Skip to main content

K–Ar age constraints on the sources of K minerals in bentonites of the Ankara-Çankırı Basin, Central Anatolia, Turkey

Abstract

Many of the bentonite deposits of the Ankara-Çankırı Basin, Central Anatolia, Turkey were found within the Miocene Hancılı Formation, which comprises lacustrine sedimentary and volcaniclastic rocks that interfinger with Uludere pyroclastic rocks of the Miocene Galatean volcanic province. In the present study, the conventional K–Ar geochronological method was used to evaluate the contribution of volcanic materials to the K-bearing components of the bentonite clay fractions. Four dacite samples from near the southern end of the basin were indistinguishable in K–Ar age (average 18.4 Ma, standard deviation 0.3 Ma). K–Ar measurements of feldspar-enriched rock fragments and hydrobiotite separated from andesitic tuff from near the northern end of the basin indicated an age of 17 ± 1 Ma. The K–Ar age values of clay fractions of bentonites, which ranged from 77 ± 5 Ma to 162 ± 5 Ma, indicate that most of the K in the bentonite clay fractions occurs in minerals derived from the Mesozoic basement rocks adjacent to the Miocene basin. The K–Ar age values support field observations indicating that these bentonites are secondary bentonites formed by alteration of volcanic components during or after deposition of volcaniclastic phases. The K-bearing mineral components of these clay fractions consisted mostly of unaltered illitic material of detrital origin whereas the smectitic components were formed by alteration of Miocene pyroclastic materials.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. Adiyaman Ö, Chorowicz J, Arnaud ON, Gündogdu MN, Gourgaud A (2001) Late Cenozoic tectonics and volcanism along the North Anatolian Fault: new structural and geochemical data. Tectonophysics 338:135–165. https://doi.org/10.1016/S0040-1951(01)00131-7

    Article  Google Scholar 

  2. Baadsgaard H, Lerbekmo JF, McDougall I (1988) A radiometric age for the Cretaceous-Tertiary boundary based upon K–Ar, Rb-Sr, and U-Pb ages of bentonites from Alberta, Saskatchewan, and Montana. Can J Earth Sci 25:1088–1097. https://doi.org/10.1139/e88-106

    Article  Google Scholar 

  3. Bakır S, Akbulut A, Kapkaç F, Karahan DS, Çetin C (2012) Türkiye Bentonit Envanteri (Envanter Serisi 204), Maden Tetkik ve Arama Genel Müdürlüğü, Ankara [Turkey Bentonite Inventory (Inventory Series 204), General Directorate of Mineral Research and Exploration, Ankara]

  4. Bigazzi G, Yeğingil Z, Ercan T, Oddone M, Özdoğan M (1993) Fission track dating obsidians in Central and Northern Anatolia. Bull Volcanol 55:588–595. https://doi.org/10.1007/BF00301811

    Article  Google Scholar 

  5. Bozkaya Ö, Günal-Türkmenoğlu A, Göncüoğlu MC, Ünlüce Ö, Yilmaz İÖ, Schroeder PA (2016) Illitization of late devonian-early carboniferous K-bentonites from Western Pontides, NW Turkey: implications for their origin and age. Appl Clay Sci 134:257–274. https://doi.org/10.1016/j.clay.2016.08.020

    Article  Google Scholar 

  6. Clauer N, Honty M, Fallick AE, Šucha V, Aubert A (2014) Regional illitization in bentonite beds from the East Slovak Basin based on isotopic characteristics (K–Ar, δ18O and δD) of illite-type nanoparticles. Clay Miner 49:247–275. https://doi.org/10.1180/claymin.2014.049.2.07

    Article  Google Scholar 

  7. Cohen KM, Finney SC, Gibbard PL, Fan J-X (2013) The ICS international chronostratigraphic chart. Episodes 36:199–204

    Article  Google Scholar 

  8. Dalrymple GB, Lanphere MA (1969) Potassium-argon dating, principles, techniques and applications to geochronology. WH Freeman and Company, San Francisco

    Google Scholar 

  9. Dangerfield A, Harris R, Sarıfakıoğlu E, Dilek Y (2011) Tectonic evolution of the Ankara Mélange and associated Eldivan ophiolite near Hançili, central Turkey. In: Wakabayashi J, Dilek Y (eds) Mélanges: processes of formation and societal significance. Geological Society of America Special Paper 480, pp 143–169. https://doi.org/10.1130/2011.2480(06)

  10. Dönmez M, Akçay AE (2010) Geological map of Çankırı H 30 quadrangle, scale 1:100.000. General Directorate of Mineral Research and Exploration (MTA) Publications, Ankara, Turkey

    Google Scholar 

  11. Elliott WC (1993) Origin of the Mg-smectite at the Cretaceous/Tertiary (K/T) boundary at Stevns Klint, Denmark. Clays Clay Miner 41:442–452. https://doi.org/10.1346/CCMN.1993.0410405

    Article  Google Scholar 

  12. Elliott WC, Aronson JL (1987) Alleghenian episode of K-bentonite illitization in the southern Appalachian Basin. Geology 15:735–739. https://doi.org/10.1130/0091-7613(1987)15%3c735:AEOKII%3e2.0.CO;2

    Article  Google Scholar 

  13. Elliott WC, Aronson JL (1993) The timing and extent of illite formation in Ordovician K-bentonites at the Cincinnati Arch, the Nashville Dome and north-eastern Illinois Basin. Basin Res 5:125–135. https://doi.org/10.1111/j.1365-2117.1993.tb00061.x

    Article  Google Scholar 

  14. Elliott WC, Aronson JL, Millard HT Jr, Gierlowski-Kordesch E (1989) The origin of the clay minerals at the Cretaceous/Tertiary boundary in Denmark. Geol Soc Am Bull 101:702–710. https://doi.org/10.1130/0016-7606(1989)101%3c0702:TOOTCM%3e2.3.CO;2

    Article  Google Scholar 

  15. Elliott WC, Edenfield AM, Wampler JM, Matisoff G, Long PE (1999) The kinetics of the smectite to illite transformation in Cretaceous bentonites, Cerro Negro, New Mexico. Clays Clay Miner 47:286–296. https://doi.org/10.1346/CCMN.1999.0470304

    Article  Google Scholar 

  16. Engels JC, Ingamells CO (1977) Geostandards—a new approach to their production and use. Geostandards Newsl 1:51–60. https://doi.org/10.1111/j.1751-908X.1977.tb00858.x

    Article  Google Scholar 

  17. Ercan T, Yegingil Z, Bigazzi G., Öddöne M, Özdoğan M (1990) Kuzeybatı Anadolu obsidiyen buluntularının kaynak belirleme çalışmaları. [Source determination studies of obsidian finds in northwestern Anatolia.] Jeoloji Mühendisliği Dergisi [Journal of Geological Engineering] 36:19–32

  18. Gradstein FM, Ogg JG, Smith AG (eds) (2005) A geologic time scale 2004. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511536045

    Book  Google Scholar 

  19. Hoffman J, Hower J, Aronson JL (1976) Radiometric dating of time of thrusting in the disturbed belt of Montana. Geology 4:16–20. https://doi.org/10.1130/0091-7613(1976)4%3c16:RDOTOT%3e2.0.CO;2

    Article  Google Scholar 

  20. Huff WD, Anderson TB, Rundle CC, Odin GS (1991) Chemostratigraphy, K–Ar ages and illitization of Silurian K-bentonites from the Central Belt of the Southern Uplands-Down-Longford terrane, British Isles. J Geol Soc 148:861–868. https://doi.org/10.1144/gsjgs.148.5.0861

    Article  Google Scholar 

  21. Jackson ML (1979) soil chemical analyses—advanced course, 2nd edn. M.L. Jackson, Madison

    Google Scholar 

  22. Jeans CV, Merriman RJ, Mitchell JG (1977) Origin of middle jurassic and lower cretaceous fuller’s earths in England. Clay Miner 12:11–44. https://doi.org/10.1180/claymin.1977.012.1.02

    Article  Google Scholar 

  23. Jeans CV, Merriman RJ, Mitchell JG, Bland DJ (1982) Volcanic clays in the Cretaceous of southern England and northern Ireland. Clay Miner 17:105–156. https://doi.org/10.1180/claymin.1982.017.1.10

    Article  Google Scholar 

  24. Jeans CV, Wray DS, Merriman RJ, Fisher MJ (2000) Volcanogenic clays in Jurassic and Cretaceous strata of England and the North Sea Basin. Clay Miner 35:25–55. https://doi.org/10.1180/000985500546710

    Article  Google Scholar 

  25. Kadir S, Külah T, Önalgil N, Erkoyun H, Elliott WC (2017) Mineralogy, geochemistry, and genesis of bentonites in Miocene volcanic-sedimentary units of the Ankara-Çankırı Basin, central Anatolia, Turkey. Clays Clay Miner 65:64–91. https://doi.org/10.1346/CCMN.2017.064051

    Article  Google Scholar 

  26. Karadenizli L (2011) Oligocene to Pliocene palaeogeographic evolution of the Çankırı-Çorum Basin, central Anatolia, Turkey. Sediment Geol 237:1–29. https://doi.org/10.1016/j.sedgeo.2011.01.008

    Article  Google Scholar 

  27. Kaymakci N (2000) Tectono-stratigraphical evolution of the Çankırı Basin (Central Anatolia Turkey). Thesis, Utrecht University. Geologica Ultraiectina, Issue 190:1–247

  28. Keller J, Jung D, Eckhardt F-J, Kreuzer H (1992) Radiometric ages and chemical characterization of the Galatean andesite massif, Pontus, Turkey. Acta Vulcanol 2:267–276

    Google Scholar 

  29. Kunk MJ, Sutter JF (1984) 40Ar/39Ar age spectrum dating of biotite from Middle Ordovician bentonites, eastern North America. In: Bruton DL (ed) Aspects of the Ordovician System. Palaeontological Contributions from the University of Oslo, No. 295, Universitetsforlaget, pp 11–22

  30. Kunze GW, Dixon JB (1986) Pretreatment for mineralogical analysis. In: Klute A (ed) Methods of Soil Analysis, Part 1. Physical and Mineralogical Methods, 2nd edn. American Society of Agronomy, Inc. and the Soil Science Society of America, Inc, Madison, pp 91–100

    Google Scholar 

  31. McCarty DK, Sakharov BA, Drits VA (2009) New insights into smectite illitization: a zoned K-bentonite revisited. Am Mineral 94:1653–1671. https://doi.org/10.2138/am.2009.3260

    Article  Google Scholar 

  32. Min K, Renne PR, Huff WD (2001) 40Ar/39Ar dating of Ordovician K-bentonites in Laurentia and Baltoscandia. Earth Planet Sci Lett 185:121–134. https://doi.org/10.1016/S0012-821X(00)00365-4

    Article  Google Scholar 

  33. Moore DM, Reynolds RC (1997) X-ray diffraction and the identification and analysis of clay minerals, 2nd edn. Oxford University Press, New York

    Google Scholar 

  34. Odin GS et al (1982) Interlaboratory standards for dating purposes. In: Odin GS (ed) Numerical dating in stratigraphy. Wiley, New York, pp 123–150

    Google Scholar 

  35. Okay Aİ, Göncüoğlu MC (2004) The Karakaya Complex: a review of data and concepts. Turk J Earth Sci 13:77–95

    Google Scholar 

  36. Osborn SG, Duffield LT, Elliott WC, Wampler JM, Elmore RD, Engel MH (2014) The timing of diagenesis and thermal maturation of the Cretaceous Marias River Shale, Disturbed Belt, Montana. Clays Clay Miner 62:112–125. https://doi.org/10.1346/CCMN.2014.0620204

    Article  Google Scholar 

  37. Samson SD, Patchett PJ, Roddick JC, Parrish RR (1989) Origin and tectonic setting of Ordovician bentonites in North America: isotopic and age constraints. Geol Soc Am Bull 101:1175–1181. https://doi.org/10.1130/0016-7606(1989)101%3c1175:OATSOO%3e2.3.CO;2

    Article  Google Scholar 

  38. Sarıfakıoğlu E, Dilek Y, Sevin M (2017) New synthesis of the Izmir-Ankara-Erzincan suture zone and the Ankara mélange in northern Anatolia based on new geochemical and geochronological constraints. In: Sorkhabi R (ed) Tectonic evolution, collision, and seismicity of Southwest Asia: in honor of Manuel Berberian’s forty-five years of research contributions. Geological Society of America Special Paper 525. https://doi.org/10.1130/2017.2525(19)

  39. Sevin M, Uğuz MF (2011) Geological map of Çankırı G 30 quadrangle, scale 1:100.000. General Directorate of Mineral Research and Exploration (MTA) Publications, Ankara, Turkey

    Google Scholar 

  40. Środoń J, Clauer N, Huff W, Dudek T, Banaś M (2009) K–Ar dating of the Lower Palaeozoic K-bentonites from the Baltic Basin and the Baltic Shield: implications for the role of temperature and time in the illitization of smectite. Clay Miner 44:361–387. https://doi.org/10.1180/claymin.2009.044.3.361

    Article  Google Scholar 

  41. Steiger RH, Jӓger E (1977) Subcommission on Geochronology: convention on the use of decay constants in geo- and cosmochronology. Earth Planet Sci Lett 36:359–362. https://doi.org/10.1016/0012-821X(77)90060-7

    Article  Google Scholar 

  42. Toprak V, Savascin Y, Gülec N, Tankut A (1996) Structure of the Galatean Volcanic Province, Turkey. Int Geol Rev 38:747–758. https://doi.org/10.1080/00206819709465358

    Article  Google Scholar 

  43. Türkecan A, Hepşen N, Papak İ, Akbaş B,Dinçel A, Karataş S, Özgür İB, Akay E, Bedi Y, Sevin M, Mutlu G, Sevin D, Ünay E, Saraç G (1991) Seben-Gerede (Bolu) - Güdül-Beypazarı (Ankara) ve Çerkeş-Orta-Kurşunlu (Çankırı) yörelerinin (Köroğlu Dağları) jeolojisi ve volkanik kayaçların petrolojisi. [Geology and petrology of volcanic rocks of Seben-Gerede (Bolu)—Güdül-Beypazarı (Ankara) and Çerkeş-Orta-Kurşunlu (Çankırı) regions (Köroğlu Mountains)]. MTA Report, No: 9193, General Directorate of Mineral Research and Exploration (MTA) Publications, Ankara, Turkey

  44. Türkmenoğlu A, Aker S (1990) Origin of sedimentary bentonite deposits of Çankiri basin, Turkey. In: Farmer VC, Tardy Y (eds) Proceedings of the 9th international clay conference, Strasbourg, 1989, Vol IV: clays in sediments, diagenesis and hydrothermalism, (Sciences Géologiques. Mémoire 88). Institut de Géologie—Université Louis-Pasteur, Strasbourg, pp 63–72. https://www.persee.fr/doc/sgeol_0302-2684_1990_act_88_1_2170

  45. Wagner GA, Weiner KL (1987) Spaltspurenanalyse an obsidianproben. [Fission track analysis on obsidian samples.] In: Korfmann M (ed) Demircihüyük: Die Ergebnisse der Ausgrabungen 1975–1978, Band II: Naturwissenschaftliche Untersuchungen [Demircihüyük: the results of the excavations, Vol 2, Scientific Investigations]. Philipp von Zabern, Mainz am Rhein, pp 24–29

  46. Warr LN, Hofmann H, van der Pluijm BA (2017) Constraining the alteration history of a Late Cretaceous Patagonian volcaniclastic bentonite–ash–mudstone sequence using K–Ar and 40Ar/39Ar isotopes. Int J Earth Sci 106:255–268. https://doi.org/10.1007/s00531-016-1315-2

    Article  Google Scholar 

  47. Weaver CE (1963) Interpretative value of heavy minerals from bentonites. J Sediment Res 33:343–349. https://doi.org/10.1306/74D70E4D-2B21-11D7-8648000102C1865D

    Article  Google Scholar 

  48. Whitney DL, Evans BW (2010) Abbreviations for names of rock-forming minerals. Am Mineral 95:185–187. https://doi.org/10.2138/am.2010.3371

    Article  Google Scholar 

  49. Wilson M, Tankut A, Guleç N (1997) Tertiary volcanism of the Galatia province, north-west Central Anatolia, Turkey. Lithos 42:105–121. https://doi.org/10.1016/S0024-4937(97)00039-X

    Article  Google Scholar 

Download references

Acknowledgments

This present study was supported financially by the Scientific Research Projects Fund of Eskişehir Osmangazi University in the framework of Projects 2014–656. Use of the K–Ar facility at Georgia State University, USA, was supported with funds from the Faculty International Partnership Engagement (FIPE) Grant ID # FIPE-16-446 to W. Crawford Elliott in cooperation with Selahattin Kadir.

Author information

Affiliations

Authors

Corresponding author

Correspondence to W. Crawford Elliott.

Appendix

Appendix

Derivation of the K–Ar age value of pre-Miocene clay in a bentonite clay fraction as a function of the proportion of the clay-fraction K that is in Miocene or younger clay

Definitions

Bentonite clay fraction—the part of a bentonite sample separated as grains smaller than 2 µm.

Component 1—those parts of a bentonite clay fraction formed after the beginning of the Miocene Period.

Component 2—those parts of a bentonite clay fraction formed before the beginning of the Miocene Period.

40Ar*—radiogenic argon; also used, by convention, for the amount (amount of substance) of radiogenic argon in a material.

40K—the radioactive isotope of terrestrial K; also used, by convention, for the amount (amount of substance) of 40K in a material.

T—the K–Ar age value of a material.

Λ—the decay constant of 40K.

λε—the partial decay constant for the transformation of 40K to 40Ar (almost entirely by electron capture).

λβ—the partial decay constant for the transformation of 40K to 40Ca by β-decay.

Kc—the amount (amount of substance) of potassium in a bentonite clay fraction.

K1—in a bentonite clay fraction, the amount of potassium from component 1.

K2—in a bentonite clay fraction, the amount of potassium from component 2.

Derivation

From a general relationship given by Dalrymple and Lanphere (1969, p. 48),

$${}^{40}{\text{Ar}}^{*} = {}^{40}{\text{K}}\frac{{\lambda_{\varepsilon } }}{{\lambda_{\varepsilon } + \lambda_{\beta } }}\left( {{\text{e}}^{{(\lambda_{\varepsilon } + \lambda_{\beta } )t}} - 1} \right),$$
(1a)

three specific relationships may be written,

$${}^{40}{\text{Ar}}_{\text{c}}^{ *} = {}^{40}{\text{K}}_{\text{c}} \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right),$$
(1b)
$${}^{40}{\text{Ar}}_{1}^{*} = {}^{40}{\text{K}}_{1} \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right),$$
(1c)

And

$${}^{40}{\text{Ar}}_{2}^{*} = {}^{40}{\text{K}}_{2} \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right),$$
(1d)

where the subscripts c, 1, and 2 denote the entire clay fraction and components 1 and 2, respectively, and the sum of the partial decay constants, λε and λβ, has been replaced by the total decay constant λ.

By definition,

$${}_{{}}^{40} {\text{Ar}}_{{ {\text{c}}}}^{*} = {}_{{}}^{40} {\text{Ar}}_{ 1}^{*} + {}_{{}}^{40} {\text{Ar}}_{ 2}^{*} .$$
(2)

Substituting in (2) the expressions for 40Ar*1 and 40Ar*2 from (1c) and (1d),

$${}^{40}{\text{Ar}}_{\text{c}}^{*} = {}^{40}{\text{K}}_{1} \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) + {}^{40}{\text{K}}_{2} \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(3)

Dividing (3) by 40Kc,

$$\frac{{{}^{ 4 0}{\text{Ar}}_{{ {\text{c}}}}^{ *} }}{{{}^{ 4 0}{\text{K}}_{\text{c}} }} = \frac{{{}^{ 4 0}{\text{K}}_{ 1} }}{{{}^{ 4 0}{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) + \frac{{{}^{ 4 0}{\text{K}}_{ 2} }}{{{}^{ 4 0}{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(4)

By convention, 40K/K in natural terrestrial materials is considered equal to 0.001167 (Steiger and Jäger 1977), so a ratio of 40K amounts in two different natural materials is the same as the ratio of K amounts in those two materials. It follows from (4) that

$$\frac{{{}^{ 4 0}{\text{Ar}}_{\text{c}}^{ *} }}{{{}^{ 4 0}{\text{K}}_{\text{c}} }} = \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) + \frac{{{\text{K}}_{ 2} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right),$$
(5)

where the subscripts c, 1, and 2 continue to denote the entire clay fraction and components 1 and 2, respectively.

Dividing (1b) by 40Kc,

$$\frac{{{}^{ 4 0}{\text{Ar}}_{\text{c}}^{ *} }}{{{}^{ 4 0}{\text{K}}_{\text{c}} }} = \frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right).$$
(6)

Equating the two expressions for 40Arc*/40Kc in (5) and (6),

$$\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right) = \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) + \frac{{{\text{K}}_{ 2} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(7)

By definition,

$${\text{K}}_{1} + {\text{K}}_{2} = {\text{K}}_{\text{c}} ,$$
(8)

so by subtraction of K1,

$${\text{K}}_{ 2} = {\text{K}}_{\text{c}} - {\text{K}}_{1}$$
(9)

Dividing (9) by Kc,

$$\frac{{{\text{K}}_{ 2} }}{{{\text{K}}_{\text{c}} }} = 1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}.$$
(10)

Replacing \(\frac{{{\text{K}}_{ 2} }}{{{\text{K}}_{\text{c}} }}\) in (7) with its equivalent from (10),

$$\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right) = \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) + \left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(11)

Subtracting the first right-hand term from (11),

$$\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{c} }} - 1} \right) - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right) = \left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(12)

Dividing (12) by \(\left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }\),

$$\frac{{\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{c} }} - 1} \right) - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right)}}{{\left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }}} = \left( {{\text{e}}^{{\lambda t_{2} }} - 1} \right).$$
(13)

Adding 1 to (13),

$$\frac{{\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right) - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right)}}{{\left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }}} + 1 = {\text{e}}^{{\lambda t_{2} }} .$$
(14)

Taking the natural logarithms,

$$\ln \left\{ {\frac{{\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right) - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right)}}{{\left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }}} + 1} \right\} = \lambda t_{2} .$$
(15)

Dividing (15) by λ and reversing,

$$t_{2} = \frac{1}{\lambda }\ln \left\{ {\frac{{\frac{{\lambda_{\varepsilon} }}{\lambda }\left( {{\text{e}}^{{\lambda t_{\text{c}} }} - 1} \right) - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}\frac{{\lambda_{\varepsilon } }}{\lambda }\left( {{\text{e}}^{{\lambda t_{1} }} - 1} \right)}}{{\left( {1 - \frac{{{\text{K}}_{ 1} }}{{{\text{K}}_{\text{c}} }}} \right)\frac{{\lambda_{\varepsilon } }}{\lambda }}} + 1} \right\}.$$
(16)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Elliott, W.C., Wampler, J.M., Kadir, S. et al. K–Ar age constraints on the sources of K minerals in bentonites of the Ankara-Çankırı Basin, Central Anatolia, Turkey. Int J Earth Sci (Geol Rundsch) 109, 2353–2367 (2020). https://doi.org/10.1007/s00531-020-01904-x

Download citation

Keywords

  • Ankara-Çankırı Basin
  • Bentonite
  • K–Ar age
  • Volcaniclastics
  • Turkey