Two copper cylinders, immersed in a water tank at 46.3°C contain helium and nitrogen, respectively. The helium-filled cylinder has a volume twice as large as the nitrogen-filled cylinder.

a) Calculate the average kinetic energy of a helium molecule at 46.3°C.

b) Calculate the average kinetic energy of a nitrogen molecule at 46.3°C.

c) Determine the molar specific heat at constant volume (CV) and at constant pressure (Cp) for helium. Enter CV first. Give answer in J/mol·K.

d) Determine the molar specific heat at constant volume (CV) and at constant pressure (Cp) for nitrogen. Enter CV first. Give answer in J/mol·K.

e) Find γ for helium.

f) Find γ for nitrogen.

To answer these questions, we will need to use certain formulas and equations. Let's break down each question and explain the steps to find the answers.

a) Calculate the average kinetic energy of a helium molecule at 46.3°C.

The average kinetic energy of a gas molecule can be calculated using the equation:

KE_avg = (3/2) * k * T,

where KE_avg is the average kinetic energy, k is the Boltzmann constant (1.38 x 10^-23 J/K), and T is the temperature in Kelvin.

To convert the given temperature from Celsius to Kelvin, we need to add 273.15.

So, the first step is to convert the given temperature to Kelvin:

T = 46.3°C + 273.15 = 319.45 K

Now, we can substitute the values in the equation:

KE_avg (helium) = (3/2) * (1.38 x 10^-23 J/K) * (319.45 K)

Simplify the equation to find the value of KE_avg (helium).

b) Calculate the average kinetic energy of a nitrogen molecule at 46.3°C.

Follow the exact same steps as in part (a), but this time we will use the molar mass of nitrogen (28 g/mol) instead of helium (4 g/mol) in the calculations. The molar mass is necessary because it affects the average kinetic energy of the gas molecules.

c) Determine the molar specific heat at constant volume (CV) and at constant pressure (Cp) for helium.

The molar specific heat at constant volume (CV) can be calculated using the equation:

CV = (f/2) * R,

where CV is the molar specific heat at constant volume, f is the degree of freedom (for a diatomic gas like helium, f = 5), and R is the gas constant (8.314 J/mol·K).

Multiply the values in the equation to find the value of CV (helium).

d) Determine the molar specific heat at constant volume (CV) and at constant pressure (Cp) for nitrogen.

Follow the same steps as in part (c), but this time use the relevant degree of freedom for nitrogen (f = 5) and the molar mass of nitrogen (28 g/mol) to calculate CV (nitrogen).

e) Find γ for helium.

The ratio of the molar specific heat at constant pressure (Cp) to the molar specific heat at constant volume (CV) is denoted by the symbol γ (gamma). It can be calculated using the equation:

γ = Cp / CV.

Use the values of CV (helium) and Cp (helium) calculated in parts (c) and (d), respectively, to find the value of γ (helium).

f) Find γ for nitrogen.

Follow the same steps as in part (e), but use the values of CV (nitrogen) and Cp (nitrogen) to calculate γ (nitrogen).

By following these steps and using the relevant formulas, you should be able to find the answers to all the questions.