What is the ratio of specific heats for hydrogen ? Given density of hydrogen at STP = 0.09 kg/m`""^3`. Velocity of sound in hydrogen at STP= 1255 m/s.

Answer

To find the ratio of specific heats (γ) for hydrogen, we can use the formula for the velocity of sound in a gas, which is given by:

\[ V = \sqrt{\frac{\gamma P}{\rho}} \]

Where:
- \( V \) is the velocity of sound,
- \( \gamma \) is the ratio of specific heats (Cp/Cv),
- \( P \) is the pressure,
- \( \rho \) is the density of the gas.

### Step 1: Write down the known values
- Density of hydrogen (\( \rho \)) = 0.09 kg/m³
- Velocity of sound in hydrogen (\( V \)) = 1255 m/s
- Pressure at STP (\( P \)) = \( 10^5 \) N/m²

### Step 2: Rearrange the formula to solve for γ
Starting from the formula:

\[ V = \sqrt{\frac{\gamma P}{\rho}} \]

We can square both sides to eliminate the square root:

\[ V^2 = \frac{\gamma P}{\rho} \]

Now, rearranging for γ gives us:

\[ \gamma = \frac{V^2 \cdot \rho}{P} \]

### Step 3: Substitute the known values into the formula
Now we can substitute the known values into the equation:

\[ \gamma = \frac{(1255)^2 \cdot 0.09}{10^5} \]

### Step 4: Calculate \( V^2 \)
First, calculate \( V^2 \):

\[ V^2 = 1255^2 = 1576025 \, \text{m}^2/\text{s}^2 \]

### Step 5: Substitute and calculate γ
Now substitute \( V^2 \) back into the equation for γ:

\[ \gamma = \frac{1576025 \cdot 0.09}{10^5} \]

Calculating the numerator:

\[ 1576025 \cdot 0.09 = 141841.25 \]

Now divide by \( 10^5 \):

\[ \gamma = \frac{141841.25}{100000} = 1.4184125 \]

### Step 6: Round the result
Finally, rounding to two decimal places gives:

\[ \gamma \approx 1.42 \]

### Conclusion
Thus, the ratio of specific heats for hydrogen is approximately:

\[ \gamma \approx 1.42 \]

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