To find the heat capacity at constant pressure (C_p) of the gas, we can follow these steps:
### Step 1: Understand the relationship between speed of sound, specific heats, and gas properties
The speed of sound in a gas is given by the formula:
\[
V = \sqrt{\frac{\gamma R T}{M}}
\]
where:
- \( V \) = speed of sound
- \( \gamma \) = ratio of specific heats (\( \frac{C_p}{C_v} \))
- \( R \) = universal gas constant (\( 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \))
- \( T \) = temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol
### Step 2: Convert the mass of the gas to moles
Given that the gas has a mass of \( 4.0 \, \text{g} \) and occupies \( 22.4 \, \text{L} \) at NTP, we can calculate the number of moles (n) using the molar volume at NTP (22.4 L/mol):
\[
n = \frac{4.0 \, \text{g}}{22.4 \, \text{g/mol}} = 0.1786 \, \text{mol}
\]
However, since we are given the specific heat capacity per mole, we can directly use the mass in kg for calculations.
### Step 3: Calculate the molar mass of the gas
Since \( 22.4 \, \text{L} \) corresponds to \( 1 \, \text{mol} \) at NTP, the molar mass \( M \) can be calculated as:
\[
M = \frac{4.0 \, \text{g}}{1 \, \text{mol}} = 0.004 \, \text{kg/mol}
\]
### Step 4: Calculate gamma (\( \gamma \))
Rearranging the speed of sound formula to solve for \( \gamma \):
\[
\gamma = \frac{M V^2}{R T}
\]
Substituting the known values:
- \( M = 0.004 \, \text{kg/mol} \)
- \( V = 952 \, \text{m/s} \)
- \( R = 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \)
- \( T = 273 \, \text{K} \)
Calculating \( \gamma \):
\[
\gamma = \frac{0.004 \times (952)^2}{8.314 \times 273}
\]
Calculating the numerator:
\[
0.004 \times 905904 = 3623.616
\]
Calculating the denominator:
\[
8.314 \times 273 = 2270.622
\]
Now, calculating \( \gamma \):
\[
\gamma = \frac{3623.616}{2270.622} \approx 1.597 \approx 1.6
\]
### Step 5: Calculate the heat capacity at constant pressure (\( C_p \))
Using the relationship between \( C_p \) and \( C_v \):
\[
C_p = \gamma C_v
\]
Given \( C_v = 5.0 \, \text{J K}^{-1} \text{mol}^{-1} \):
\[
C_p = 1.6 \times 5.0 = 8.0 \, \text{J K}^{-1} \text{mol}^{-1}
\]
### Final Answer
The heat capacity at constant pressure \( C_p \) is:
\[
\boxed{8.0 \, \text{J K}^{-1} \text{mol}^{-1}}
\]
