Question

For the equilibrium,
\( 2\text{A}_{(g)} \rightleftharpoons 2\text{B}_{(g)} + 2\text{C} \)
the value of \( K_p \) is 0.1662 atm at 1000 K. The value of \( K_c \) at the same temperature is ( \( R = 0.0831 \) )

Hint:

In competitive exams, if solving a problem strictly as written leads to messy, random numbers while a slight, plausible typo correction leads to elegant, perfectly canceling numbers, the corrected path is almost always the intended one. Pay attention to unit clues too!

Answer 1

Step 1: Use Relation Between \( K_p \) and \( K_c \)
\[ K_p = K_c (RT)^{\Delta n} \] \[ K_c = \frac{K_p}{(RT)^{\Delta n}} \]

Step 2: Calculate \( \Delta n \)
\[ \Delta n = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants}) \] \[ \Delta n = (2 + 2) - 2 = 2 \]

Step 3: Substitute Given Values
\[ K_p = 0.1662,\quad R = 0.0831,\quad T = 1000\,\text{K} \] \[ RT = 0.0831 \times 1000 = 83.1 \]

Step 4: Calculate \( K_c \)
\[ K_c = \frac{0.1662}{(83.1)^2} \] \[ K_c = \frac{0.1662}{6905.61} \] \[ K_c \approx 2.4 \times 10^{-5} \]

Step 5: Final Answer
\[ \boxed{K_c = 2.4 \times 10^{-5}} \]