The equilibrium constant in terms of pressure `(K_(p))` and concentration `(K_(c ))` are related as `(Delta n` is change in number of gas moles)

Options

A. `K_(p)=K_(c ) xx (RT)^(1-Deltan)`
B. `K_(p)=((K_(c))/(RT))^(Deltan)`
C. `K_(p)=K_(c ) (RT)^(Delta n)`
D. `K_(p)=((RT)^(Deltan))/(K_(c))`

Explanation

To derive the relationship between the equilibrium constants \( K_p \) and \( K_c \), we can follow these steps:

### Step 1: Understand the Definitions
- \( K_p \) is the equilibrium constant expressed in terms of partial pressures of the gases.
- \( K_c \) is the equilibrium constant expressed in terms of molar concentrations of the gases.

### Step 2: Write the General Relationship
The relationship between \( K_p \) and \( K_c \) can be expressed as:
\[ K_p = K_c \cdot R^n \cdot T^{\Delta n} \]
where:
- \( R \) is the universal gas constant (0.0821 L·atm/(K·mol)),
- \( T \) is the temperature in Kelvin,
- \( \Delta n \) is the change in the number of moles of gas, calculated as the moles of gaseous products minus the moles of gaseous reactants.

### Step 3: Calculate \( \Delta n \)
To find \( \Delta n \), we need to analyze a specific reaction. For example, consider the reaction:
\[ aA + bB \rightleftharpoons cC + dD \]
Here, \( \Delta n \) can be calculated as:
\[ \Delta n = (c + d) - (a + b) \]
This represents the difference between the total moles of products and the total moles of reactants.

### Step 4: Substitute \( \Delta n \) into the Equation
Once \( \Delta n \) is calculated, substitute it back into the equation:
\[ K_p = K_c \cdot R^{\Delta n} \cdot T^{\Delta n} \]

### Step 5: Example Calculation
For a specific example, consider the reaction:
\[ A + B \rightleftharpoons 2C + D \]
Here, the number of moles of products is \( 2 + 1 = 3 \) and the number of moles of reactants is \( 1 + 1 = 2 \). Thus:
\[ \Delta n = 3 - 2 = 1 \]
Now, substituting \( \Delta n \) into the relationship gives:
\[ K_p = K_c \cdot R^1 \cdot T^1 \]
or simply:
\[ K_p = K_c \cdot R \cdot T \]

### Conclusion
The relationship between \( K_p \) and \( K_c \) is established, and we can use this to convert between the two constants depending on the conditions of the reaction.

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