To determine which of the given options is correct for the reaction [tex]\( CO + \frac{1}{2} O_2 \rightleftharpoons CO_2 \)[/tex], we need to understand the relationship between the equilibrium constants [tex]\( K_p \)[/tex] and [tex]\( K_c \)[/tex]. This relationship is governed by the equation:
[tex]\[ K_p = K_c(RT)^{\Delta n} \][/tex]
where:
- [tex]\( R \)[/tex] is the universal gas constant (0.0821 L·atm/(K·mol))
- [tex]\( T \)[/tex] is the temperature in Kelvin
- [tex]\( \Delta n \)[/tex] is the change in moles of gas, calculated as (moles of gaseous products) - (moles of gaseous reactants)
For the given reaction:
[tex]\[ CO + \frac{1}{2} O_2 \rightleftharpoons CO_2 \][/tex]
Firstly, calculate [tex]\( \Delta n \)[/tex]:
- Moles of gaseous products: 1 mole (from [tex]\( CO_2 \)[/tex])
- Moles of gaseous reactants: 1 mole (from [tex]\( CO \)[/tex]) + 0.5 mole (from [tex]\( \frac{1}{2} O_2 \)[/tex]) = 1.5 moles
Thus:
[tex]\[ \Delta n = 1 - 1.5 = -0.5 \][/tex]
Next, since [tex]\( \Delta n \)[/tex] is negative, we need to evaluate the term [tex]\( (RT)^{\Delta n} \)[/tex]:
- Given [tex]\( R = 0.0821 \)[/tex] L·atm/(K·mol)
- Assuming temperature [tex]\( T = 298 \)[/tex] K (standard room temperature)
We calculate [tex]\( (RT)^{\Delta n} \)[/tex]:
[tex]\[ (RT)^{\Delta n} = (0.0821 \times 298)^{-0.5} \][/tex]
[tex]\[ \approx 0.2022 \][/tex]
Given that [tex]\( (RT)^{\Delta n} \approx 0.2022 \)[/tex] is less than 1, this implies:
[tex]\[ K_p < K_c \][/tex]
Therefore, the correct option is:
(c) [tex]\( K_p < K_c \)[/tex]
