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 the elementary reaction 2NO + O₂ → 2NO₂ when the volume of the reaction vessel is halved, what is the change in the rate of reaction?

It becomes eight times

For the elementary reaction 2NO + O₂ → 2NO₂ 

r = k [NO]²[O₂]¹ 

If volume of the vessel is halved, concentration of each reactant becomes doubled.

r' = k[2NO]²[2O₂]¹ 

r' = 8r