To determine in which of the following equations [tex]\( K_p = K_c \)[/tex], we have to understand the relationship between these two equilibrium constants. The relation between [tex]\( K_p \)[/tex] and [tex]\( K_c \)[/tex] is given by the equation:
[tex]\[ K_p = K_c (RT)^{\Delta n} \][/tex]
where:
- [tex]\( R \)[/tex] is the universal gas constant,
- [tex]\( T \)[/tex] is the temperature in Kelvin,
- [tex]\( \Delta n \)[/tex] is the change in the number of moles of gas between the reactants and the products.
For [tex]\( K_p \)[/tex] to equal [tex]\( K_c \)[/tex], the exponent [tex]\( \Delta n \)[/tex] must be zero. This would make [tex]\( (RT)^{\Delta n} \)[/tex] equal to 1, since any number raised to the power of zero is 1.
Now, let's evaluate [tex]\( \Delta n \)[/tex] for each given equation:
### Option A: [tex]\( 2H_{2(g)} + C_2H_{2(g)} \rightleftharpoons C_2H_6(g) \)[/tex]
- Reactants: 2 moles of [tex]\( H_2(g) \)[/tex] + 1 mole of [tex]\( C_2H_2(g) \)[/tex] [tex]\( \rightarrow \)[/tex] Total = 3 moles
- Products: 1 mole of [tex]\( C_2H_6(g) \)[/tex]
- [tex]\( \Delta n \)[/tex]= moles of products - moles of reactants = [tex]\( 1 - 3 = -2 \)[/tex]
### Option B: [tex]\( 2NO_{(g)} + O_2_{(g)} \rightleftharpoons 2NO_2_{(g)} \)[/tex]
- Reactants: 2 moles of [tex]\( NO(g) \)[/tex] + 1 mole of [tex]\( O_2(g) \)[/tex] [tex]\( \rightarrow \)[/tex] Total = 3 moles
- Products: 2 moles of [tex]\( NO_2(g) \)[/tex]
- [tex]\( \Delta n \)[/tex]= moles of products - moles of reactants = [tex]\( 2 - 3 = -1 \)[/tex]
### Option C: [tex]\( N_2_{(g)} + O_2_{(g)} \rightleftharpoons 2NO_{(g)} \)[/tex]
- Reactants: 1 mole of [tex]\( N_2(g) \)[/tex] + 1 mole of [tex]\( O_2(g) \)[/tex] [tex]\( \rightarrow \)[/tex] Total = 2 moles
- Products: 2 moles of [tex]\( NO(g) \)[/tex]
- [tex]\( \Delta n \)[/tex]= moles of products - moles of reactants = [tex]\( 2 - 2 = 0 \)[/tex]
### Option D: [tex]\( 2H_2_{(g)} + O_2_{(g)} \rightleftharpoons 2H_2O_{(g)} \)[/tex]
- Reactants: 2 moles of [tex]\( H_2(g) \)[/tex] + 1 mole of [tex]\( O_2(g) \)[/tex] [tex]\( \rightarrow \)[/tex] Total = 3 moles
- Products: 2 moles of [tex]\( H_2O(g) \)[/tex]
- [tex]\( \Delta n \)[/tex]= moles of products - moles of reactants = [tex]\( 2 - 3 = -1 \)[/tex]
The correct answer is:
C. [tex]\( N_2_{(g)} + O_2_{(g)} \rightleftharpoons 2NO_{(g)} \)[/tex]
In this case, [tex]\( \Delta n = 0 \)[/tex], which means that [tex]\( (RT)^{\Delta n} = 1 \)[/tex]. Therefore, [tex]\( K_p = K_c \)[/tex].
