Question 258: Thermodynamics & KTG

Question

The total kinetic energy of 1 mole of oxygen at $$27^{\circ} \mathrm{C}$$ is :
[Use universal gas constant $$(R)=8.31 \mathrm{~J} /$$ mole K]

Accepted Answer

The total kinetic energy of a mole of an ideal gas can be determined by using the equipartition theorem, which states that the energy is equally distributed among degrees of freedom. For a diatomic molecule such as oxygen ($$O_2$$), there are 5 degrees of freedom (3 translational and 2 rotational - assuming the vibrational modes are not excited at room temperature), so each degree of freedom has an average energy of $$\frac{1}{2} kT$$ per molecule, where $$k$$ is the Boltzmann constant and $$T$$ is the temperature in kelvins.
However, since we are dealing with moles, we'll use the universal gas constant $$R$$ instead of the Boltzmann constant $$k$$, because $$R = k \cdot N_A$$ where $$N_A$$ is the Avogadro constant (the number of molecules in a mole). Therefore, the average energy per mole for each degree of freedom is $$\frac{1}{2}RT$$.
To find the total energy, we multiply the energy per degree of freedom by the number of degrees of freedom for the diatomic gas:
$$ E_{	ext{total}} = 	ext{degrees of freedom} 	imes \frac{1}{2} R T $$
For diatomic oxygen:
$$ E_{	ext{total}} = 5 	imes \frac{1}{2} R T $$
Given that temperature $$ T $$ is $$ 27^{\circ} \mathrm{C} $$, we first convert it to kelvins:
$$ T_{	ext{K}} = T_{	ext{C}} + 273.15 = 27 + 273 = 300 	ext{ K} $$
Now we plug in the values for $$ R $$ and $$ T_{	ext{K}} $$:
$$ E_{	ext{total}} = 5 	imes \frac{1}{2} 	imes 8.31 	ext{ J/mol} \cdot 	ext{K} 	imes 300 	ext{ K} $$
When we calculate this, we find:
$$ E_{	ext{total}} = \frac{5}{2} 	imes 8.31 	imes 300 $$
$$ E_{	ext{total}} = 6232.5 	ext{ J/mol} $$
So the total kinetic energy of 1 mole of oxygen at $$27^{\circ} \mathrm{C}$$ is approximately 6232.5 J.

Answer

6232.5 J

Answer

5670.5 J

Answer

6845.5 J

Answer

5942.0 J

Thermodynamics & KTG

The total kinetic energy of 1 mole of oxygen at $27^{\circ} \mathrm{C} is : [Use universal gas constant (R)=8.31 \mathrm{~J} /$ mole K]

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