Question

Assuming you have a thermometer and a device to measure the speed of sound in a gas, explain how you can determine the mole fraction of helium in a mixture of helium gas and air.

Short answer

Answer: Follow these steps to determine the mole fraction of helium: 1. Write down the values for the adiabatic index and molar masses of helium and air. 2. Measure the temperature (in Kelvin) and speed of sound in the gas mixture. 3. Calculate the effective molar mass of the gas mixture using the measured values and assuming γ of about 1.4. 4. Determine the mole fraction of helium (x_He) using the following formula: \[x_{He} = \frac{M_{air} - M_{effective}}{M_{air} - M_{He}}\] where M_air is the molar mass of air, M_He is the molar mass of helium, and M_effective is the effective molar mass of the mixture.

Step-by-step solution

Understanding the speed of sound in a gas

First, we need to understand the relationship between the speed of sound in a gas, denoted by c, and other properties of the gas. The speed of sound in a gas is given by the following formula: \[c = \sqrt{\frac{\gamma RT}{M}}\] where: - c is the speed of sound - γ (gamma) is the adiabatic index or ratio of specific heats (Cp/Cv) - R is the ideal gas constant (8.314 J/(mol K)) - T is the temperature of the gas in Kelvin - M is the molar mass of the gas in kg/mol Note that the adiabatic index depends on the specific gas, so we will need to know the values for both helium and air.

Obtain the adiabatic index and molar masses for helium and air

For helium, the adiabatic index (γ) is approximately 5/3 (1.67), and the molar mass (M) is approximately 0.004 kg/mol. For dry air, the adiabatic index is approximately 7/5 (1.4), and the molar mass is approximately 0.029 kg/mol. These values will be used in the calculation of the mole fraction of helium in the gas mixture.

Measure Temperature and Speed of Sound

Using the thermometer, measure the temperature of the gas mixture in Kelvin. Then, use the device to measure the speed of sound in the gas mixture.

Calculate the effective molar mass of the gas mixture

Using the measured values for the temperature and the speed of sound in the gas mixture, determine the effective molar mass (M_effective) of the mixture. This can be calculated using the speed of sound equation mentioned in Step 1: \[M_{effective} = \frac{\gamma R T}{c^2}\] We still do not know the value of γ for the mixture. Instead, we can assume that the mixture of gases behaves as a monoatomic gas, given that helium is monoatomic and air is primarily composed of diatomic gases. We can assume a value for γ of about 1.4. Compute the effective molar mass using the measured temperature and speed of sound.

Determine the mole fraction of helium in the gas mixture

We will now determine the mole fraction of helium (x_He) in the mixture using the following equation: \[x_{He} = \frac{M_{air} - M_{effective}}{M_{air} - M_{He}}\] where: - x_He is the mole fraction of helium in the mixture - M_air is the molar mass of air (0.029 kg/mol) - M_He is the molar mass of helium (0.004 kg/mol) - M_effective is the effective molar mass of the mixture (calculated in Step 4) Compute the mole fraction of helium using the obtained values, and you will have the desired result for this exercise.