During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of \(\frac{C_p}{C_v}\) for the gas is :
- \(\frac{5}{3}\)
- \(\frac{3}{2}\)
- \(\frac{7}{5}\)
- \(\frac{9}{7}\)
The Correct Option is B
Approach Solution - 1
To solve this problem, we need to understand the relationship between pressure \(P\), temperature \(T\), and the specific heat ratio \(\gamma = \frac{C_p}{C_v}\) during an adiabatic process. Given that the pressure of the gas is proportional to the cube of its absolute temperature, we can express this relationship as:
\(P \propto T^3\)
For an adiabatic process, the relation between pressure and temperature is given by:
\(PT^{-\frac{\gamma}{\gamma - 1}} = \text{constant}\)
Since \(P \propto T^3\), we can equate the exponents to get:
\(T^3 \cdot T^{-\frac{\gamma}{\gamma - 1}} = \text{constant}\)
\(3 - \frac{\gamma}{\gamma - 1} = 0\)
Solving the above equation:
\(3 = \frac{\gamma}{\gamma - 1}\)
\(3(\gamma - 1) = \gamma\)
\(3\gamma - 3 = \gamma\)
\(2\gamma = 3\)
\(\gamma = \frac{3}{2}\)
This gives us the specific heat ratio \(\frac{C_p}{C_v} = \gamma = \frac{3}{2}\).
Therefore, the correct answer is:
\(\frac{3}{2}\)
Explanation of Options:
- The option \(\frac{5}{3}\) corresponds to diatomic gases with additional energy degrees. Our derivation does not support this value for the given condition.
- The option \(\frac{3}{2}\) was derived based on the given condition and is correct.
- The option \(\frac{7}{5}\) is typically related to diatomic gases as well, which don't match the condition stated.
- The option \(\frac{9}{7}\) does not fit the temperature-proportional relationship provided.
Approach Solution -2
Given:
\(P \propto T^3 \implies P T^{-3} = \text{constant}.\)
From the adiabatic relation:
\(P V^\gamma = \text{constant}.\)
Using the ideal gas law:
\(P \left(\frac{nRT}{P}\right)^\gamma = \text{constant}.\)
Simplify:
\(P^{1-\gamma} T^\gamma = \text{constant}.\)
Substitute \( P \propto T^3 \):
\(P^{1-\gamma} T^\gamma = T^3 \implies P^{1-\gamma} T^{\gamma - 3} = \text{constant}.\)
Reorganize to find the relationship between \( \gamma \) and the exponents:
\(P^{1-\gamma} T^{\gamma - 3} = \text{constant}.\)
Equating powers of \(T\):
\(\frac{\gamma}{1 - \gamma} = -3.\)
Solve for \( \gamma \):
\(\gamma = -3 + 3\gamma.\)
Simplify:
\(3 = 2\gamma \implies \gamma = \frac{3}{2}.\)
The Correct answer is: \(\frac{3}{2}\)
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