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.
a) To calculate the average kinetic energy of a helium molecule at 46.3°C, we can use the formula:
K.E. = (3/2) * k * T
Where K.E. is the average kinetic energy, k is the Boltzmann constant (1.38 * 10^-23 J/K), and T is the temperature in Kelvin.
First, let's convert the temperature from Celsius to Kelvin:
T = 46.3 + 273.15 = 319.45 K
Now we can calculate the average kinetic energy of a helium molecule:
K.E. = (3/2) * (1.38 * 10^-23 J/K) * (319.45 K)
K.E. ≈ 3.47 * 10^-21 J
b) To calculate the average kinetic energy of a nitrogen molecule at 46.3°C, we follow the same formula:
K.E. = (3/2) * k * T
Using the same temperature of 319.45 K from part (a), we can calculate the average kinetic energy of a nitrogen molecule:
K.E. = (3/2) * (1.38 * 10^-23 J/K) * (319.45 K)
K.E. ≈ 3.47 * 10^-21 J
c) The molar specific heat at constant volume (CV) for a gas can be estimated using the formula:
CV = (f/2) * R
Where f is the degree of freedom of the gas molecule (for a monoatomic gas like helium, it is 3), and R is the gas constant (8.314 J/mol·K).
So, for helium:
CV = (3/2) * 8.314 J/mol·K
CV ≈ 12.47 J/mol·K
d) Similarly, for nitrogen:
CV = (f/2) * R
For a diatomic gas like nitrogen, the degree of freedom is 5 (3 translational + 2 rotational).
CV = (5/2) * 8.314 J/mol·K
CV ≈ 20.785 J/mol·K
e) The ratio of specific heat capacities (γ) for helium can be calculated using the formula:
γ = CP / CV
Since we have calculated CV in part (c), we need to calculate CP now. For an ideal gas, the relation between CP and CV is:
CP = CV + R
CP = 12.47 J/mol·K + 8.314 J/mol·K
CP ≈ 20.784 J/mol·K
Now we can calculate γ for helium:
γ = 20.784 J/mol·K / 12.47 J/mol·K
γ ≈ 1.665
f) Similarly, for nitrogen, we calculate CP using the formula:
CP = CV + R
CP = 20.785 J/mol·K + 8.314 J/mol·K
CP ≈ 29.099 J/mol·K
Now we can calculate γ for nitrogen:
γ = 29.099 J/mol·K / 20.785 J/mol·K
γ ≈ 1.401