The heat of reaction for an endothermic reaction at constant volume in equilibrium is 1200 cal more than al constant pressure at 300K, then

Options

A. `Delta_((g)) = - 2`
B. `(Kp)/(Kc) = 1.648 xx 10^(-3)`
C. `Delta n_((g)) = + 2 `
D. `(Kc)/(Kp) = 1.648 xx 10^(-3)`

Answer

To solve the problem step-by-step, we will analyze the information given and apply the relevant thermodynamic equations.

### Step 1: Understand the relationship between ΔU and ΔH
For a reaction, the relationship between the change in internal energy (ΔU) and the change in enthalpy (ΔH) at constant volume and constant pressure is given by the equation:
\[ \Delta H = \Delta U + \Delta nRT \]
Where:
- ΔH = change in enthalpy
- ΔU = change in internal energy
- Δn = change in the number of moles of gas
- R = universal gas constant (approximately 2 cal/(K·mol) for this context)
- T = temperature in Kelvin

### Step 2: Set up the equation based on the problem statement
According to the problem, the change in internal energy at constant volume is 1200 cal more than the change in enthalpy at constant pressure:
\[ \Delta U - \Delta H = 1200 \text{ cal} \]
Rearranging gives:
\[ \Delta U = \Delta H + 1200 \text{ cal} \]

### Step 3: Substitute ΔU in the enthalpy equation
Substituting ΔU from the rearranged equation into the enthalpy equation:
\[ \Delta H = (\Delta H + 1200) + \Delta nRT \]
This simplifies to:
\[ 0 = 1200 + \Delta nRT \]
Rearranging gives:
\[ \Delta nRT = -1200 \]

### Step 4: Solve for Δn
Now, we can isolate Δn:
\[ \Delta n = \frac{-1200}{RT} \]
Substituting R = 2 cal/(K·mol) and T = 300 K:
\[ \Delta n = \frac{-1200}{2 \times 300} = \frac{-1200}{600} = -2 \]

### Step 5: Relate Δn to Kp and Kc
Using the relationship between Kp and Kc:
\[ K_p = K_c (RT)^{\Delta n} \]
Substituting Δn = -2, R = 0.0821 L·atm/(K·mol), and T = 300 K:
\[ K_p = K_c (0.0821 \times 300)^{-2} \]

### Step 6: Calculate the value
Calculating the constant:
\[ 0.0821 \times 300 = 24.63 \]
Thus:
\[ K_p = K_c (24.63)^{-2} = K_c \cdot \frac{1}{605.0569} \approx K_c \cdot 0.00165 \]

### Step 7: Conclusion
The ratio \( \frac{K_p}{K_c} \) is approximately \( 1.648 \times 10^{-3} \). Therefore, the correct option is the one that corresponds to this value.