3 slight clean up
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$K_{c}$$K_\mathrm c$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}\begin{equation} K_\mathrm c = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$$K_\mathrm c$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}\begin{equation} \text{unit of }K_\mathrm c = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

$K_{c}$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

$K_\mathrm c$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_\mathrm c = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_\mathrm c$ has the unit

\begin{equation} \text{unit of }K_\mathrm c = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

2 replaced http://chemistry.stackexchange.com/ with https://chemistry.stackexchange.com/
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$K_{c}$ is defined as (for more information see herehere)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

$K_{c}$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

$K_{c}$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}

1
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$K_{c}$ is defined as (for more information see here)

\begin{equation} K_{c} = \prod_i [c_{i}]^{\nu_{i}} = \frac{\prod \limits_{i \in \text{products}} [c_{i}]^{|\nu_{i}|}}{\prod \limits_{j \in \text{reactants}} [c_{j}]^{|\nu_{j}|}} \end{equation}

where $\nu_{i}$ and $c_{i}$ are the stochiometric coefficient and the concentration of the $i^{\text{th}}$ component in the reaction, respectively.

If you use this definition for the reaction

\begin{equation} \ce{2SO2 + O2 -> 2SO3} \end{equation}

you get

\begin{equation} K_{c} = \frac{[\ce{SO3}]^2}{[\ce{SO2}]^{2} [\ce{O2}]} \end{equation}

where the notation $[i] = c_{i}$ was used. This equation tells you that $K_{c}$ has the unit

\begin{equation} \text{unit of }K_{c} = \frac{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2}}{\left(\frac{\mathrm{mol}}{\mathrm{L}} \right)^{2} \left(\frac{\mathrm{mol}}{\mathrm{L}} \right)} = \frac{\mathrm{L}}{\mathrm{mol}} \ . \end{equation}