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Changes in which factors affect both the rate and the rate constant of a first order reaction?

I. temperature

II. concentration

(A) I only (B) II only (C) Both I and II (D) Neither I nor II

I would think that the answer would be (C), both temperature and concentration.

Concentration would obviously affect it [from the rate expression], and the temperature would also affect it because the arrhenius equation for the rate constant is $$k = A\cdot\exp\left\{\frac{-E^*}{RT}\right\}$$

and thus temperature would have an influence on the rate constant, and since the rate constant is part of the rate equation, it would also affect the rate of the reaction itself.

However, the given answer is A. I don't understand why this is.

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If you see the rate equation for the Arrhenius equation: $$ k = A e^{-E_a/(R T)} $$ you can see that the rate constant increases for an increase in temperature (and as activation energy increases, the rate constant decreases.

Now, since the rate constant increases, this implies that the rate of reaction increases with temperature.

Regarding concentration: you are correct in that concentration affects the rate of reaction. However, a rate constant does not change according to concentration. You can observe this in the Arrhenius equation, where A is independent of the concentration of the substrate. The fact that the rate increases does not imply the rate constant changes.

Possibly more clearly: it is implied that increasing the rate constant will increase the rate of reaction. However, increasing the rate does not increase the rate constant.

An increase in temperature increases the rate constant and hence the rate.

An increase in concentration increases the rate but not the rate constant.

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the rate of a reaction depends on the rate constant and concentration R=k[A][B].. the rate constant according to Arrhenius is as u stated. from ur equation u can see that concentration doesnt affect k but affects R. Temperature affects k and k affects R so, temperature affects both while concentration affects only Rate of reactions.

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It is necessary to distinguish between rate and rate constant (also called rate coefficient). The rate of a second order reaction A+B$\rightarrow$C is expressed as $$d[A]/dt = -k_2[A][B]$$ where $k_2$ is the rate constant. (The negative sign is because A is consumed in the reaction). In the case of a first order reaction, e.g. a cis-trans isomerisation, the rate is $$d[A]/dt=-k_1[A]$$ and the rate constant $k_1$.
As you can see from the Arrhenius equation the rate constant ($k_1, k_2$) depends on temperature and the rate depends on concentration of both A and B and the rate constant. So temperature affects rate and rate constant, and concentration only affects rate. So the answer is (A)

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