Process undergone by a gas sample is described in terms of thermodynamic temperature (T) and volume occupied by gas (V) by the equation ${TV}^{x} = constant.$ Degrees of freedom of the gas sample is f. Choose the correct statement(s).

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For $x = 2,$ the process represents adiabatic process

For $x = 1,$ temperature of the gas sample should decrease if heat is supplied to the gas

For $x = - 1$ , Bulk modulus of the gas sample is zero

For $x = - 1$ , coefficient of cubical expansion of gas is inverse of thermodynamic temperature

answer is B, C, D.

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Detailed Solution

$(1) T V^{3 - 1} = c o n s \tan t . S o i f t h e p r o c e s s i s a d i a b a t i c, γ = 3 w h i c h y i e l d s f = 1 w h i c h i s n o t p o s s i b l e f o r a n y t y p e o f g a s . (2) T V = c o n s \tan t . W h e n h e a t i s s u p p l i e d t o t h e g a s, V i n c r e a s e s a n d T d e c r e a s e s . (3) T V^{- 1} = c o n s \tan t \Rightarrow T = V \Rightarrow \frac{1}{V} \frac{d T}{d V} = \frac{1}{T} (4) T V^{- 1} = c o n s \tan t \Rightarrow P = c o n s \tan t \Rightarrow B = V \frac{d P}{d V} = 0$