The reaction, $CO(g) + 3{H_2}(g) \leftrightarrow C{H_4}(g) + {H_2}O$ , is at equilibrium at 1300 K in a 1L flask. It also contain 0.30 mol of CO, 0.10 mol of ${H_2}$​ and 0.02 mol of ${H_2}O$ and an unknown amount of $C{H_4}$​ in the flask. Determine the concentration of $C{H_4}$​ in the mixture. The equilibrium constant, ${K_c}$ for the reaction at the given temperature is 3.90.

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Hint: ${K_p}$ and ${K_c}$ are the signs used to represent the equilibrium constant of an ideal gaseous mixture. “${K_p}$” is an equilibrium constant which is used to denote equilibrium concentrations expressed in terms of atmospheric pressure. While “${K_c}$” is an equilibrium constant used when equilibrium concentrations are expressed in terms of molarity. ${K_c}$ is the ratio of the equilibrium concentrations of the products over the equilibrium concentrations of the reactants each raised to the power of their stoichiometric coefficients.

Complete Step by Step Solution:
Let the concentration of $C{H_4}$(methane) at equilibrium be x.
$CO(g) + 3{H_2}(g) \leftrightarrow C{H_4}(g) + {H_2}O$
At equilibrium , $\dfrac{{0.3}}{1} = 0.3M$ $\dfrac{{0.1}}{1} = 0.1M$ $x$ $\dfrac{{0.2}}{1} = 0.2M$
It is given that ${K_c}$ = 3.90.
Therefore,
Therefore,
$\dfrac{{[C{H_4}(g)][{H_2}O]}}{{[CO(g)][{H_2}(g)]}} = {K_c} \\$
$\Rightarrow \dfrac{{x \times 0.02}}{{0.3 \times {{(0.1)}^3}}} = 3.90 \\$
$\Rightarrow x = \dfrac{{3.90 \times 0.3 \times {{(0.1)}^3}}}{{0.02}} \\$
=$\dfrac{{0.00117}}{{0.02}}$
= 0.0585 M
=$5.85 \times {10^{ - 2}}$M
Therefore, the concentration of $C{H_4}$ at equilibrium is $5.85 \times {10^{ - 2}}$ M.

Additional information: Molarity is one of the most widely used unit of concentration. It is denoted by the sign “M”. It is defined as no. of moles of solute that is present in 1L of solution. An ideal gas is a hypothetical/imaginary gas that is made up of a group of randomly propagating very tiny/point particles that intermix only in the event of elastic collisions.

Note: The equilibrium constant of a chemical reaction which is denoted by the symbol K, provides information regarding the relationship that exists between the products and reactants when a chemical reaction reaches equilibrium. For example, the equilibrium constant of a chemical reaction at equilibrium can be defined as the ratio of the concentration of the product side of the reaction to the concentration of the reactant compounds of the equation. Both the reactants and products are raised to their respective stoichiometric coefficients. It is an important note that there are several different types of equilibrium constants that provide relationships between the products and the reactants of equilibrium reactions in terms of different units.