A certain chemical reaction has an activation energy of 0.500 × 10−19 J. What temperature is required for the reaction rate to be double what it is at 20°C?

To answer this question, we need to use the Arrhenius equation which describes the temperature dependence of reaction rates. The equation is:

k = Ae^(-Ea/RT)

where: k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature (in Kelvin).

The problem states that the reaction rate doubles when the temperature increases. This means that k2/k1 = 2, where k1 is the rate constant at T1 (20°C or 293.15K) and k2 is the rate constant at T2 (the temperature we want to find).

We can set up the equation as follows:

k2/k1 = Ae^(-Ea/RT2) / Ae^(-Ea/RT1) = e^(Ea/R * (1/T1 - 1/T2)) = 2

Taking the natural logarithm of both sides gives us:

Ea/R * (1/T1 - 1/T2) = ln(2)

We can solve this equation for T2:

1/T2 = 1/T1 - ln(2)/(Ea/R)

Substituting the given values (Ea = 0.500 × 10−19 J, R = 8.314 J/(mol*K), T1 = 293.15K) gives us:

1/T2 = 1/293.15 - ln(2)/(0.500 × 10−19 / 8.314)

Solving this equation will give us the value of T2 in Kelvin. To convert it to Celsius, subtract 273.15 from the result.