To calculate the equilibrium constant \( K_p \) for the reaction given that \( K_c = 0.77 \) at a temperature of 570 K, we can use the relationship between \( K_p \) and \( K_c \), which is expressed by the formula:

\[ K_p = K_c \cdot R \cdot T^{\Delta n} \]

In this equation:

• \( R \) is the ideal gas constant, \( 0.08206 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \)
• \( T \) is the temperature in Kelvin, which is given as 570 K
• \( \Delta n \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

For the reaction:

2 A(g) + 1 B(s) ⇌ C(g) + 3 E(g)

We identify the moles of gas in the products: 1 (from C) + 3 (from E) = 4 moles. The moles of gas in the reactants consist only of A, which has 2 moles. Therefore:

\[ \Delta n = 4 - 2 = 2 \]

Now, substituting the known values into the equation:

\[ K_p = 0.77 \cdot 0.08206 \cdot 570^{2} \]

Calculating \( 570^{2} \):

\[ 570^{2} = 324900 \]

Next, we calculate \( 0.08206 \cdot 324900 \):

\[ 0.08206 \cdot 324900 \approx 26564.07494 \]

Now, multiplying this result by \( K_c \):

\[ K_p = 0.77 \cdot 26564.07494 \approx 20453.93 \]

Finally, rounding to the appropriate number of significant figures (2 significant figures, based on \( K_c \) and temperature), we express the final answer as:

\[ K_p \approx 2.0 \times 10^{4} \]

This value represents the equilibrium constant \( K_p \) for the reaction at the specified temperature.