Ethyl chloride vapor decomposes by the first-order reaction: C2H5Cl -> C2H4 + HCl. The activation energy is 249 kJ/mol, and the frequency factor is 1.6 * 10^14 s^-1. Find the temperature at which the rate of the reaction would be twice as fast.

Verified step by step guidance

1. Identify the Arrhenius equation: $k = A e^{-\frac{E_a}{RT}}$, where $k$ is the rate constant, $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the temperature in Kelvin.
2. Recognize that the problem asks for the temperature at which the rate of the reaction is twice as fast, meaning $k_2 = 2k_1$.
3. Use the Arrhenius equation to set up the relationship between the two rate constants: $\frac{k_2}{k_1} = \frac{A e^{-\frac{E_a}{RT_2}}}{A e^{-\frac{E_a}{RT_1}}} = 2$.
4. Simplify the equation: $e^{-\frac{E_a}{RT_2}} = 2 e^{-\frac{E_a}{RT_1}}$ and take the natural logarithm of both sides to solve for $T_2$: $-\frac{E_a}{RT_2} = \ln(2) - \frac{E_a}{RT_1}$.
5. Rearrange the equation to solve for $T_2$: $T_2 = \frac{E_a}{R(\ln(2) + \frac{E_a}{RT_1})}$, where $E_a = 249 \text{ kJ/mol}$, $R = 8.314 \text{ J/mol K}$, and $T_1$ is the initial temperature.

Key Concepts

First-Order Reactions

First-order reactions are chemical reactions where the rate is directly proportional to the concentration of one reactant. This means that if the concentration of the reactant doubles, the rate of the reaction also doubles. The rate law for a first-order reaction can be expressed as rate = k[A], where k is the rate constant and [A] is the concentration of the reactant.

Arrhenius Equation

The Arrhenius equation describes how the rate constant of a reaction depends on temperature and activation energy. It is given by k = A * e^(-Ea/RT), where k is the rate constant, A is the frequency factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. This equation helps in understanding how temperature influences reaction rates.

Temperature and Reaction Rate

Temperature plays a crucial role in chemical reaction rates, as an increase in temperature typically leads to an increase in reaction rate. This is due to the higher kinetic energy of molecules, which results in more frequent and effective collisions. In the context of the Arrhenius equation, a higher temperature decreases the exponent's value, thereby increasing the rate constant k, which can lead to a doubling of the reaction rate at a specific temperature.