11 grade physics others

The molar specific heats of an ideal gas at constant pressure and volume are denoted by Cp and Cv, respectively. If γ = Cp/Cv and R is the universal gas constant, then Cv is equal to:
(A) (1 + γ) / (1 - γ)
(B) R / (γ - 1)
(C) (γ - 1) / R
(D) γR

Aniket Singh

1 Year agoGrade

1 Answer

Askiitians Tutor Team

1 Year ago

To solve for CvC_v, the molar specific heat of an ideal gas at constant volume, we use the following relationships for an ideal gas:
1. Relationship between CpC_p and CvC_v:
Cp=Cv+RC_p = C_v + R
where RR is the universal gas constant.
2. Definition of γ\gamma:
γ=CpCv\gamma = \frac{C_p}{C_v}
Using these two relationships, we can express CvC_v in terms of γ\gamma and RR.
Step 1: Substituting CpC_p into the equation for γ\gamma
γ=Cv+RCv\gamma = \frac{C_v + R}{C_v}
Step 2: Simplifying the equation
Multiply through by CvC_v:
γCv=Cv+R\gamma C_v = C_v + R
Rearranging:
Cv(γ−1)=RC_v (\gamma - 1) = R
Step 3: Solving for CvC_v
Cv=Rγ−1C_v = \frac{R}{\gamma - 1}
Final Answer:
The value of CvC_v is:
Cv=Rγ−1C_v = \frac{R}{\gamma - 1}
The correct option is: (B) Rγ−1\frac{R}{\gamma - 1}.