A 2.60-mol sample of oxygen gas is confined to a 5.03-L vessel at a pressure of 7.95 atm. Find the average translational kinetic energy of the oxygen molecules under these conditions.

To find the average translational kinetic energy of the oxygen molecules, we can use the formula:

Average translational kinetic energy (KE) = (3/2) * (R * T),

where R is the ideal gas constant and T is the temperature in Kelvin.

Step 1: Convert given values to the required units.
The pressure is given in atm, but we need to use the SI unit of pressure, which is Pascal (Pa).
1 atm = 101325 Pa
So, the pressure (P) = 7.95 atm * 101325 Pa/atm = 805668.75 Pa

Step 2: Determine the temperature.
The ideal gas law can be used to find the temperature, using the equation:
PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

Rearrange the equation to solve for T:
T = (PV) / (nR)

Plug in the given values and solve:
T = (805668.75 Pa * 5.03 L) / (2.60 mol * 8.314 J/(mol*K))
T ≈ 952.18 K

Step 3: Calculate the average translational kinetic energy.
Using the formula:
KE = (3/2) * (R * T)

Plug in the values for R and T and solve:
KE = (3/2) * (8.314 J/(mol*K) * 952.18 K)
KE ≈ 11895.29 J/mol

So, the average translational kinetic energy of the oxygen molecules under these conditions is approximately 11895.29 J/mol.

To find the average translational kinetic energy of the oxygen molecules, we can use the kinetic theory of gases, which states that the average translational kinetic energy (KE) of a gas is directly proportional to its temperature.

First, we need to find the temperature (T) of the oxygen gas sample. We can use the ideal gas law equation: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

To start, let's convert the pressure from atm to Pascal, the volume from L to m^3, and the temperature from Celsius to Kelvin.

Given:
Number of moles (n) = 2.60 mol
Volume (V) = 5.03 L = 0.00503 m^3
Pressure (P) = 7.95 atm = 805197.5 Pa (1 atm = 101325 Pa)
Ideal gas constant (R) = 8.314 J/(mol*K)

Now, rearrange the ideal gas law equation to solve for temperature (T):
T = (PV) / (nR)

Substitute the given values:
T = (805197.5 Pa * 0.00503 m^3) / (2.60 mol * 8.314 J/(mol*K))

T = 0.195 K

Now that we have the temperature in Kelvin (K), we can calculate the average translational kinetic energy using the formula:

KE_avg = (3/2) * k * T

where k is Boltzmann's constant, approximately 1.38 x 10^-23 J/K.

Substitute the values:
KE_avg = (3/2) * (1.38 x 10^-23 J/K) * (0.195 K)

KE_avg = 1.27 x 10^-23 J

Therefore, the average translational kinetic energy of the oxygen molecules under these conditions is approximately 1.27 x 10^-23 Joules.