Argon gas at 305 K is confined within a constant volume at a pressure p. If the gas has a pressure `p_2` when it is cooled to 195 K, what is the ratio of `p_2` to `p_1`?

Options

A. `0.410`
B. `0.717`
C. `0.639`
D. `1.28`

Answer

To solve the problem, we will use the ideal gas law and the relationship between pressure and temperature for a gas at constant volume.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**: The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

2. **Constant Volume Condition**: Since the volume \( V \) is constant and we are considering the same amount of gas (constant \( n \)), we can express the relationship between pressure and temperature as:
\[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \]
where \( P_1 \) and \( T_1 \) are the initial pressure and temperature, and \( P_2 \) and \( T_2 \) are the final pressure and temperature.

3. **Identify Given Values**: From the problem:
- Initial temperature \( T_1 = 305 \, \text{K} \)
- Final temperature \( T_2 = 195 \, \text{K} \)
- Initial pressure \( P_1 = p \) (unknown)
- Final pressure \( P_2 = p_2 \) (unknown)

4. **Set Up the Ratio**: Using the relationship derived from the ideal gas law:
\[ \frac{P_2}{P_1} = \frac{T_2}{T_1} \]

5. **Substitute the Values**: Now, substitute the known temperatures into the equation:
\[ \frac{P_2}{P_1} = \frac{195}{305} \]

6. **Calculate the Ratio**: Now, perform the division:
\[ \frac{P_2}{P_1} = \frac{195}{305} \approx 0.639 \]

7. **Final Result**: Thus, the ratio of \( P_2 \) to \( P_1 \) is:
\[ \frac{P_2}{P_1} \approx 0.639 \]