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: What we want.
We are asked to write the gas constant $R$ using two known things: $E$, the kinetic energy of one mole of gas, and $T$, the absolute temperature.
Step 2: The key idea from kinetic theory.
Gas molecules are always moving. Their total moving energy for one mole depends only on temperature, and the rule is \[ E = \frac{3}{2}RT \] This says hotter gas has more kinetic energy.
Step 3: Plan the algebra.
We have $E$ on the left and $R$ buried on the right. We just rearrange to get $R$ by itself.
Step 4: Clear the fraction.
Multiply both sides by 2 to remove the half: \[ 2E = 3RT \]
Step 5: Isolate R.
Now divide both sides by $3T$: \[ R = \frac{2E}{3T} \]
Step 6: Read off the answer.
This neat form matches option (2). \[ \boxed{R = \frac{2E}{3T}} \]