At 0^ ^ ^ C , the speed of sound in hydrogen is 1280 m s^ -1 . The density of hydrogen at STP is 0.089 kg m^ -3 . Determine the molar heat capacities C_p and C_V of hydrogen.

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Correct answer: A. C_p = 26.3 J K^ -1 mol^ -1 , C_V = 18.0 J K^ -1 mol^ -1

The speed of sound in a gas is given by the formula:

v = \sqrt{\dfrac{\gamma P}{\rho}}

Rearranging for the adiabatic index

\gamma

:

\gamma = \dfrac{v^2 \rho}{P}

Given:

v = 1280 \text{ m s}^{-1}

\rho = 0.089 \text{ kg m}^{-3}

At STP, taking standard pressure

P = 10^5 \text{ Pa}

(1 bar):

\gamma = \dfrac{(1280)^2 \times 0.089}{10^5} = \dfrac{1638400 \times 0.089}{100000} \approx 1.458

We know the relations for molar heat capacities:

C_p - C_V = R

\dfrac{C_p}{C_V} = \gamma

From these, we get:

C_V = \dfrac{R}{\gamma - 1}

Using

R = 8.3 \text{ J K}^{-1} \text{ mol}^{-1}

:

C_V = \dfrac{8.3}{1.458 - 1} = \dfrac{8.3}{0.458} \approx 18.1 \text{ J K}^{-1} \text{ mol}^{-1}

Now, calculating

C_p

:

C_p = C_V + R = 18.1 + 8.3 = 26.4 \text{ J K}^{-1} \text{ mol}^{-1}

These values are closest to

C_p = 26.3 \text{ J K}^{-1} \text{ mol}^{-1}

and

C_V = 18.0 \text{ J K}^{-1} \text{ mol}^{-1}

.

Answer:

C_p = 26.3 \text{ J K}^{-1} \text{ mol}^{-1}

,

C_V = 18.0 \text{ J K}^{-1} \text{ mol}^{-1}

At 0^ ^ ^ C , the speed of sound in hydrogen is 1280 m s^ -1 . The...

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