Rate of a reaction decreases with the passage of time as the concentration of reactants decrease. Conversely, rates generally increase when reactant concentrations increase. So, rate of a reaction depends upon the concentration of reactants. 

Consider a general reaction

aA + bB → cC + dD

where a, b, c and d are the stoichiometric coefficients of reactants and products.

The rate expression for this reaction is

Rate ∝ [A]x [B]y

where exponents x and y may or may not be equal to the stoichiometric coefficients (a and b) of the reactants.

Above equation can also be written as Rate = k [A]^x [B]^y

-\(\frac{dR}{dT}\) = k [A]^x [B]^y

This form of equation is known as differential rate equation, where k is a proportionality constant called rate constant.

The rate of a gaseous reaction is given by the expression k [A][B]. If volume of the reaction vessel is suddenly increased to double the initial volume, what will be the reaction rate w.r.t to the original rate?

\(\frac{1}{4}\) times

r = k[A][B]

If volume of the vessel is doubled then the concentration of the reactant gets halved.

r' = k[\(\frac{A}{2}\)][\(\frac{B}{2}\)]

r' = \(\frac{k}{4}\)[A][B]

r' = \(\frac{r}{4}\)