A chemist is working with a 4.50 mol sample of an ideal gas. The internal energy of the sample is known to be 14,900 J.
What is the average kinetic energy of each gas molecule?
The average kinetic energy of each gas molecule can be calculated using the formula:
Average kinetic energy (KE) = (3/2) * (R * T)
Where:
- KE is the average kinetic energy of each gas molecule
- R is the gas constant (8.314 J/mol·K)
- T is the temperature in Kelvin
In this case, the internal energy of the sample is given as 14,900 J.
Since internal energy (U) is equal to the average kinetic energy (KE) multiplied by the number of molecules (n), we can calculate the number of moles (n) using the formula:
n = U / KE
Substituting the given values:
4.50 mol = 14,900 J / KE
Solving for KE:
KE = 14,900 J / 4.50 mol
KE ≈ 3,311 J/mol
Now, we can calculate the average kinetic energy of each gas molecule using the formula:
KE = (3/2) * (R * T)
Rearranging the formula:
T = (2 * KE) / (3 * R)
Substituting the known values:
T = (2 * 3,311 J/mol) / (3 * 8.314 J/mol·K)
T ≈ 267.09 K
Therefore, the average kinetic energy of each gas molecule is approximately 3,311 J/mol, or 267.09 K.
To find the average kinetic energy of each gas molecule, we need to use the formula:
Average Kinetic Energy = (3/2) * (R) * (Temperature)
Where:
- Average Kinetic Energy is the average kinetic energy of each gas molecule.
- R is the ideal gas constant (8.314 J/(mol*K)).
- Temperature is the temperature of the gas in Kelvin.
In this case, we are given the internal energy of the sample, but we need to find the temperature to calculate the average kinetic energy. We can use the relationship between internal energy and temperature:
Internal Energy = (3/2) * (R) * (Number of moles) * (Temperature)
Rearranging the equation, we can solve for temperature:
Temperature = (Internal Energy) / ((3/2) * (R) * (Number of moles))
Now, let's plug in the values:
Internal Energy = 14,900 J
Number of moles = 4.50 mol
R = 8.314 J/(mol*K)
Temperature = (14,900 J) / ((3/2) * (8.314 J/(mol*K)) * (4.50 mol))
By calculating this equation, we can find the temperature in Kelvin. Then we can use the temperature in the original equation to find the average kinetic energy per molecule.