Question

A gas at pressure \( P \), volume \( V \), and temperature \( T \). If pressure is doubled and temperature is halved, the new volume is:

• A. \( V \)
• B. \( V/2 \)
• C. \( V/4 \)
• D. \( 2V \)

Hint:

When multiple variables change, track them one by one. Doubling the pressure tends to compress the gas to half its volume (\( V/2 \)). Halving the temperature compresses it even further by half again. So, \( 1/2 \times 1/2 = 1/4 \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves the behavior of ideal gases when multiple state variables change simultaneously.
The combined gas law integrates Boyle's Law, Charles's Law, and Gay-Lussac's Law into a single expression.
It states that the product of pressure and volume, divided by the absolute temperature, is constant for a fixed mass of gas.
This means that if we know the initial state of a gas (\(P_1, V_1, T_1\)) and the changes applied to any two variables, we can mathematically determine the third.
In this scenario, increasing pressure tends to compress the gas, and decreasing temperature also tends to compress it, so we should expect a significant decrease in volume.
Step 2: Key Formula or Approach:
Combined Gas Law Formula: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
Step 3: Detailed Explanation:
Initial state values: \(P_1 = P\), \(V_1 = V\), \(T_1 = T\).
Final state values: \(P_2 = 2P\) (doubled), \(T_2 = T/2\) (halved).
We need to find the new volume \(V_2\).
Substituting these into the combined gas law:
\[ \frac{P \cdot V}{T} = \frac{(2P) \cdot V_2}{T/2} \]
Simplify the right-hand side by multiplying the numerator by the reciprocal of the denominator:
\[ \frac{PV}{T} = \frac{2P \cdot V_2 \cdot 2}{T} = \frac{4P \cdot V_2}{T} \]
Cancel the common terms \(P\) and \(T\) from both sides of the equation:
\[ V = 4V_2 \implies V_2 = \frac{V}{4} \]
The volume is reduced to one-fourth of its original value.
Step 4: Final Answer:
The new volume of the gas is \(V/4\).