∆U = n × Cv × ∆T∆U = 5 × (5R/2) × (20-10)J∆U = 5×5×8.314×10/2 J∆U = 250 × 4.157 JNow 1J = 1/4.18 CalSo ∆U = 250 × 4.157 / 4.18∆U ≈ 250 cal

Change in Internal Energy of Gas

To determine the change in internal energy of the gas, we can use the equation:

ΔU = nCvΔT

Where:
ΔU is the change in internal energy
n is the number of moles of the gas
Cv is the molar heat capacity at constant volume
ΔT is the change in temperature

Gathering the Information

Given:
Number of moles of oxygen (n) = 5 moles
Change in temperature (ΔT) = 20°C - 10°C = 10°C

Understanding Molar Heat Capacity at Constant Volume (Cv)

Molar heat capacity at constant volume (Cv) is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant volume. For an ideal gas, the molar heat capacity at constant volume is given by the equation:

Cv = (f/2)R

Where:
Cv is the molar heat capacity at constant volume
f is the degree of freedom of the gas molecules
R is the ideal gas constant

For a diatomic gas like oxygen, the degree of freedom (f) is equal to 5, which includes translational motion in three dimensions and rotational motion about two axes. Therefore, for oxygen gas, f = 5.

Calculating the Change in Internal Energy

Using the equation for the change in internal energy, we can substitute the given values:

ΔU = nCvΔT
ΔU = (5 moles) * ((5/2) * R) * (10°C)

Since R is a constant, we can simplify the equation further:

ΔU = (5/2) * R * 10 moles * °C

Conversion of Temperature to Kelvin

Before calculating the final value, we need to convert the temperature from Celsius to Kelvin because the ideal gas constant (R) is typically given in units of J/(mol·K). The conversion is done by adding 273.15 to the Celsius temperature:

10°C + 273.15 = 283.15 K

Now we can substitute the value of R and calculate the change in internal energy:

ΔU = (5/2) * R * 10 moles * 283.15 K

Calculating the Final Value

To obtain the final value, we need to know the numerical value of the ideal gas constant (R), which is 8.314 J/(mol·K):

ΔU = (5/2) * 8.314 J/(mol·K) * 10 moles * 283.15 K

Simplifying the equation further, we get:

ΔU = 5873.375 J

Therefore, the change in internal energy of the gas is approximately 5873.375 J.