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Chapter 8: Problem 1098

For hydrogen gas \(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}=\mathrm{a}\) and for oxygen gas \(\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}=\mathrm{b}\), The relation between \(\mathrm{a}\) and \(\mathrm{b}\) is given by (A) \(a=4 b\) (B) \(a=b\) (C) \(\mathrm{a}=16 \mathrm{~b}\) (D) \(a=8 b\)

Short Answer

Expert verified
The relationship between a and b is given by (B) a = b, as both hydrogen and oxygen gases have the same degrees of freedom (5) and the relationship Cp - Cv equals (f/2)R for both cases, where f is the degrees of freedom and R is the gas constant. Since the degrees of freedom are the same, a and b are equal.

Step by step solution

01

Write down the given information

We have the following information: For hydrogen gas, Cp - Cv = a For oxygen gas, Cp - Cv = b We also know that Cp = Cv + R, where R is the gas constant.
02

Calculate the degrees of freedom for hydrogen and oxygen gas

Hydrogen gas is a diatomic gas, and it has 5 degrees of freedom: 3 translational + 2 rotational (vibrational modes are not considered at room temperature). Thus, for hydrogen, f1 = 5. Oxygen gas is also a diatomic gas, and it also has 5 degrees of freedom: 3 translational + 2 rotational. Therefore, for oxygen, f2 = 5.
03

Use the relationship between Cp, Cv, and the degrees of freedom

For an ideal gas, we have the relationship: Cp - Cv = (f/2)R, where f is the number of degrees of freedom. So, for hydrogen gas, we have: a = (f1/2)R = (5/2)R and for oxygen gas, we have: b = (f2/2)R = (5/2)R
04

Determine the relation between a and b

Now, we can write down the relationship between a and b by dividing the equations for hydrogen gas and oxygen gas: \( \frac{a}{b} = \frac{(5/2)R}{(5/2)R} \) On simplifying the equation, we get: \( \frac{a}{b} = 1 \) Which implies that a = b So, the correct option is: (B) a = b

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Most popular questions from this chapter

\(\mathrm{Cp}\) and Cv denote the specific heat of oxygen per unit mass at constant Pressure and volume respectively, then (A) \(\mathrm{cp}-\mathrm{cv}=(\mathrm{R} / 16)\) (B) \(\mathrm{Cp}-\mathrm{Cv}=\mathrm{R}\) (C) \(\mathrm{Cp}-\mathrm{Cv}=32 \mathrm{R}\) (D) \(\mathrm{Cp}-\mathrm{Cv}=(\mathrm{R} / 32)\)

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