Question

If $E$ is the kinetic energy per mole of an ideal gas and $T$ is the absolute temperature, then the universal gas constant ($R$) is given as

• A. $\frac{2T}{3E}$
• B. $\frac{2E}{3T}$
• C. $\frac{3T}{2E}$
• D. $\frac{3E}{2T}$

Hint:

To verify your formula dimensions quickly, remember that kinetic energy per mole $E$ has units of J/mol. Since $R$ has units of $\text{J}/(\text{mol}\cdot\text{K})$ and $T$ is in K, the expression $\frac{E}{T}$ gives the exact units of $R$. This easily eliminates options (A) and (C) where temperature is inverted in the numerator.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem asks for the expression of the universal gas constant $R$ in terms of the kinetic energy per mole $E$ and the absolute temperature $T$ of an ideal gas.

Step 2: Key Formula or Approach:
According to the kinetic theory of gases, the total translational kinetic energy per mole ($E$) of an ideal gas is directly related to its absolute temperature by the formula: $$E = \frac{3}{2}RT$$ We can rearrange this formula to isolate the universal gas constant $R$.

Step 3: Detailed Explanation:
Start with the standard thermodynamic energy equation per mole: $$E = \frac{3}{2}RT$$ To solve for $R$, multiply both sides of the equation by 2: $$2E = 3RT$$ Now, divide both sides by $3T$ to isolate the universal gas constant $R$: $$R = \frac{2E}{3T}$$ This algebraic arrangement corresponds precisely to option (B).

Step 4: Final Answer:
The universal gas constant is given by $R = \frac{2E}{3T}$, matching option (B).