Question

If the given graph shows the logarithmic values of pressure (P) and volume (V) of an ideal gas, then the ratio of the specific heat capacities of the gas is:

• A. \( 1.5 \)
• B. \( 1.2 \)
• C. \( 1.4 \)
• D. \( 1.3 \)

Hint:

For gases following \( PV^\gamma = \text{constant} \), the slope of a log-log plot of \( P \) vs. \( V \) directly gives \( -\gamma \), the ratio of specific heats.

Answer 1

The Correct Option is C

Step 1: Understanding the Graph The given graph represents the relationship between \( \log P \) and \( \log V \) for an ideal gas. Since the graph is a straight line with a negative slope, we use the equation: \[ \log P = m \log V + \text{constant} \] where \( m \) is the slope. Step 2: Calculating the Slope From the given data points on the graph: \[ m = \frac{\Delta (\log P)}{\Delta (\log V)} \] \[ m = \frac{2.48 - 2.20}{1.2 - 1.4} \] \[ m = \frac{0.28}{-0.2} = -1.4 \] Step 3: Relation to Specific Heat Ratio For an ideal gas undergoing a polytropic process: \[ PV^\gamma = \text{constant} \] where \( \gamma \) is the ratio of specific heats: \[ \gamma = -m \] Thus, \[ \gamma = 1.4 \] Conclusion Thus, the correct answer is: \[ 1.4 \]