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One mole of an ideal gas with heat capac...

One mole of an ideal gas with heat capacity `C_V` goes through a process in which its entropy `S` depends on `T` as `S = alpha//T`, where `alpha` is a constant. The gas temperature varies from `T_1` to `T_2` Find :
(a) the molar heat capacity of the gas as function of its temperature ,
(b) the amount of heat transferred to the gas ,
( c) the work performed by the gas.

Text Solution

Verified by Experts

(a) `C = T (dS)/(dT) = - (alpha)/(T)`
(b) `Q = int_(T_1)^(T_2) CdT = alpha 1n (T_1)/(T_2)`
( c) `W = Delta Q - Delta U = alpha 1n (T_1)/(T_2) + C_V (T_1 - T_2)`
Since for an ideal gas `C_V` is constant
and `Delta U = C_v(T_2 - T_1)`
(`U` does not depend on `V`).
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The molar heat capacity for an ideal gas

One mole of an ideal gas goes through a process in which the entropy of the gas changes with temperature T as S = aT + C_V 1n T , where a is a positive constant. C_V is the molar heat capacity of this gas at constant volume. Find the volume dependence of the gas temperature in this process if T = T_0 at V = V_0 .

Knowledge Check

  • Molar heat capacity for a gas at constant temperature and pressure is

    A
    `3//2` R
    B
    `5//2` R
    C
    depends on atomicity of gas
    D
    infinity `(infty)`

  • For an ideal gas the equation of a process for which the heat capacity of the gas varies with temperatue as C=(alpha//T(alpha) is a constant) is given by

    A
    `VlnT=constant`
    B
    `VT^(1//(gamma-1))_(e)^(alpha//RT)`=constant
    C
    `V^(1)/(gamma-1)T^(alpha//RT)`=constant
    D
    `V^(gamma-1)T`=constant

  • The specific heat of an ideal gas varies with temperature T as

    A
    `T^1`
    B
    `T^2`
    C
    `T^-2`
    D
    `T^0`

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