1. The half-life for the (first-order) radioactive decay of 14C is 5730 yr (it emits a β ray with an
energy of 0.16 MeV). An archaeological sample contained wood that had only 72 percent of the 14C
found in living trees. What is its age?
I got 500years?
2. The temperature dependence of the acid-catalyzed hydrolysis of penicillin was investigated,
and the dependence of the rate constant on temperature is given below. What is the activation energy
and Arrhenius preexponential factor for this hydrolysis reaction?
Temperature (oC) k (s–1)
22.2 7.0 x 10–4
27.2 9.8 x 10–4
33.7 1.6 x 10–3
38.0 2.0 x 10–3
I found it to be 3.5........?
I'm pretty lost....I'd greatly appreciate any help!
.72=e^(-.692t/5730)
t== 5730/.692 ln .72 I don't get 500 years.
2)
I don't have time to solve it now, perhaps later.
ln rate=ln A - activationenergy/RT
You will need two points to find two unknowns, knowing T and rate.
see other post
1. To find the age of the archaeological sample, we can use the concept of radioactive decay and the given information about the half-life of 14C.
First, let's understand the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay. In this case, the half-life of 14C is given as 5730 years.
Since the archaeological sample contains only 72 percent of the 14C found in living trees, it means that 28 percent of the 14C has decayed. We can assume that the remaining 72 percent corresponds to one half-life (50 percent) of the initial sample.
Now, we need to calculate how many half-lives have elapsed to reach 72 percent. We can use the formula N = N0 * (1/2)^(t/T), where N is the final amount, N0 is the initial amount, t is the time elapsed, and T is the half-life.
Since N0 = 100 percent (initial amount), N = 72 percent, and we want to find t (the age), we can rearrange the equation to solve for t:
72/100 = (1/2)^(t/5730)
To isolate the exponent (t/5730), we can take the logarithm of both sides. Let's use the natural logarithm, denoted as ln:
ln(72/100) = ln((1/2)^(t/5730))
ln(72/100) = (t/5730) * ln(1/2)
Now we can solve for t:
t/5730 = ln(72/100) / ln(1/2)
t = 5730 * (ln(72/100) / ln(1/2))
Evaluating the expression on the right-hand side, we find t ≈ 1995 years. Therefore, the age of the archaeological sample is approximately 1995 years.
2. To determine the activation energy and Arrhenius preexponential factor for the hydrolysis reaction, we can analyze the temperature dependence of the rate constant (k).
The Arrhenius equation relates the rate constant (k) to the temperature (T) and involves the activation energy (Ea) and the Arrhenius preexponential factor (A):
k = A * e^(-Ea/RT)
In this equation, R is the gas constant.
We have been given the values of the rate constant (k) at different temperatures (T). If we take the natural logarithm (ln) of both sides of the Arrhenius equation, we can transform the equation into a linear form:
ln(k) = ln(A) - (Ea/RT)
Now, we can calculate Ea and ln(A) from the given data points by performing a linear regression analysis. In this case, the temperature (T) is in degrees Celsius, but we need to convert it to Kelvin (K) because the gas constant (R) is typically given in units of J/(mol·K).
Converting the given temperatures to Kelvin:
T (K) = T (oC) + 273.15
Using this conversion, we can calculate the corresponding Kelvin temperatures for each data point and then calculate ln(k) for each data point.
Using the given data points, we have:
Temperature (oC) Temperature (K) ln(k)
22.2 295.35 -7.063
27.2 300.35 -6.927
33.7 306.85 -6.437
38.0 311.15 -6.215
Now we have a linear relationship between ln(k) and 1/T (as R is a constant):
ln(k) = ln(A) - (Ea/R) * (1/T)
Comparing this to the equation of a straight line (y = mx + b), we recognize that m corresponds to (-Ea/R) and b corresponds to ln(A).
By performing linear regression using the ln(k) values as the y-values and 1/T (in Kelvin) as the x-values, we can determine the slope (m) and the y-intercept (b).
Using the linear regression analysis on the ln(k) values and 1/T values, we can obtain the slope as (-Ea/R) and the y-intercept as ln(A).
Finally, we can calculate Ea by multiplying the slope by R and the Arrhenius preexponential factor (A) by using the y-intercept:
Ea = -m * R
A = e^b
Substituting the values obtained through linear regression, we can find the activation energy and Arrhenius preexponential factor for the hydrolysis reaction.
Note: The values you have provided for the rate constant (k) at different temperatures (T) are missing units, so it's important to ensure that the units are consistent throughout the calculations.
I hope this explanation helps you understand how to approach these questions!