Question

An ideal diatomic gas is made up of molecules that do not vibrate. Its volume compressed by a factor of 32,without any exchange of heat. If the initial and final pressures are P₁ and P₂,respectively,the ratio P₁:P₂,is:

• A. 7:5
• B. 128:1
• C. 1:32
• D. 32:1
• E. 3:08

Answer 1

The Correct Option is B

Given:

• Ideal diatomic gas with non-vibrating molecules (degrees of freedom, \( f = 5 \))
• Volume compressed by a factor of 32 (\( V_2 = V_1/32 \))
• No heat exchange (adiabatic process)
• Initial and final pressures: \( P_1 \) and \( P_2 \)

Step 1: Adiabatic Process Relation

For an adiabatic process in an ideal gas:

\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

where \( \gamma = \frac{C_p}{C_v} = \frac{f + 2}{f} \). For a diatomic gas with \( f = 5 \):

\[ \gamma = \frac{7}{5} = 1.4 \]

Step 2: Solve for Pressure Ratio

Rearrange the adiabatic relation:

\[ \frac{P_2}{P_1} = \left(\frac{V_1}{V_2}\right)^\gamma \]

Substitute \( V_2 = V_1/32 \):

\[ \frac{P_2}{P_1} = 32^\gamma = 32^{7/5} \]

Calculate \( 32^{7/5} \):

\[ 32 = 2^5 \implies 32^{7/5} = (2^5)^{7/5} = 2^7 = 128 \]

Thus:

\[ \frac{P_2}{P_1} = 128 \implies \frac{P_1}{P_2} = \frac{1}{128} \]

Conclusion:

The ratio \( P_1 : P_2 \) is 1:128.

Answer: \(\boxed{B}\)

Answer 2

To determine the ratio of the initial pressure \(P_1\) to the final pressure \(P_2\) for an ideal diatomic gas (which does not vibrate) undergoing adiabatic compression, we need to use the adiabatic condition for an ideal gas.

For an ideal diatomic gas, which has 5 degrees of freedom (3 translational + 2 rotational), the adiabatic index (γ) is:
\[ \gamma = \frac{7}{5} = 1.4 \]

The adiabatic relation for pressure and volume is given by:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

Here, \( V_1 \) is the initial volume and \( V_2 \) is the final volume. The problem states that the volume is compressed by a factor of 32, so:
\[ V_2 = \frac{V_1}{32} \]

Using the adiabatic condition, we have:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
\[ P_1 V_1^\gamma = P_2 \left( \frac{V_1}{32} \right)^\gamma \]
\[ P_1 V_1^\gamma = P_2 \frac{V_1^\gamma}{32^\gamma} \]
\[ P_1 = P_2 \frac{1}{32^\gamma} \]
\[ \frac{P_1}{P_2} = 32^\gamma \]

Substitute \(\gamma = 1.4\) into the equation:
\[ \frac{P_1}{P_2} = 32^{1.4} \]

Now we calculate \( 32^{1.4} \):
\[ 32 = 2^5 \]
\[ 32^{1.4} = (2^5)^{1.4} = 2^{5 \cdot 1.4} = 2^7 = 128 \]

Therefore, the ratio \( P_1 : P_2 \) is:
\[ P_1 : P_2 = 128 \]

So, the final answer is:
\[ P_1 : P_2 = 128 : 1 \]
Thus The correct answer is Option (B): \(128:1\)