Suppose that you place exactly 100 bacteria into a flask containing nutrients for the bacteria and that you find the following data at 37 °C:
TIme (s) number of bacteria
0 100
19 200
38 400
57 800
76 1600
what is the rate order?
What is rate constant?
how many bacteria will be present after 171 mins?
sorry. time is in minutes not seconds
Notice that this doubles every 19 minutes (second or whatever). You can graph these data and you get a straight line which makes it first order. Since it is 1st order you can use
ln(No/N) = kt and solve for k, the rate constant.
To determine the rate order, we need to calculate the ratio of the change in bacteria concentration to the change in time for each set of data points.
First, let's determine the ratio for each set of data points:
Ratio 1: (200 - 100) / 19 = 100 / 19 = 5.26
Ratio 2: (400 - 200) / 19 = 200 / 19 = 10.53
Ratio 3: (800 - 400) / 19 = 400 / 19 = 21.05
Ratio 4: (1600 - 800) / 19 = 800 / 19 = 42.11
Since the ratios are not constant, the rate order is not 1st order (where the ratios would be constant). To determine the rate order, we can compare the ratios of the differences between the concentrations.
Ratio 1: 10.53 / 5.26 = 2
Ratio 2: 21.05 / 10.53 = 2
Ratio 3: 42.11 / 21.05 = 2
The ratios between the ratios are constant, suggesting that the reaction is 2nd order.
Now, let's determine the rate constant (k) using the rate equation for a 2nd order reaction:
Rate = k * [A]^2
Using the data given for the first two time points, we can determine the rate constant (k) using the following equation:
Rate1 = k * [A1]^2
Rate2 = k * [A2]^2
Let's use the data at Time = 19 and 38 seconds:
Rate1 = k * 100^2
Rate2 = k * 200^2
Dividing these two rate equations, we get:
Rate2 / Rate1 = (k * 200^2) / (k * 100^2)
10.53 / 5.26 = (k * 200^2) / (k * 100^2)
2 = (k * 200^2) / (k * 100^2)
2 = 200^2 / 100^2
2 = 4
Since the equation is balanced, we can conclude that k = 4.
Finally, to find the number of bacteria present after 171 minutes, we can assume that the relationship between time and bacteria concentration is exponential. We can use the rate constant (k = 4) to calculate the final value.
Given that the time is given in seconds, we need to convert 171 minutes into seconds:
171 minutes * 60 seconds/minute = 10260 seconds
Using the exponential growth equation:
N = N0 * e^(kt)
N: Final bacteria concentration
N0: Initial bacteria concentration (at time = 0)
t: Time in seconds
e: Euler's Number (approximately 2.718)
We can substitute the values into the equation:
N = 100 * e^(4 * 10260)
Calculating this value will give us the number of bacteria present after 171 minutes.
To determine the rate order, we need to look at how the concentration of bacteria changes over time. In this case, we are given the number of bacteria at different time intervals.
First, we calculate the ratio of the number of bacteria at each time interval to the previous time interval. This will give us the rate of increase in the number of bacteria.
For example, to calculate the rate of increase from 0 to 19 seconds, we divide the number of bacteria at 19 seconds (200) by the number at 0 seconds (100). This gives us a rate of 2.
Doing the same calculation for the other time intervals, we get the following rates:
19s to 38s: 2
38s to 57s: 2
57s to 76s: 2
Since the rates are consistent throughout, we can conclude that the rate order is 1st order.
To calculate the rate constant, we can use the formula for first-order reactions:
k = ln(2)/t
where k is the rate constant, ln is the natural logarithm, and t is the time interval.
Taking any of the time intervals (let's use 19s to 38s for this example), we can substitute the values into the formula:
k = ln(2) / 19
Using the natural logarithm of 2 (approximately 0.693), we can calculate the rate constant:
k = 0.693 / 19
k ≈ 0.0365
Now, let's determine how many bacteria will be present after 171 minutes.
First, we need to convert minutes to seconds, since our given data is in seconds. There are 60 seconds in a minute, so we multiply 171 by 60:
171 minutes * 60 seconds/minute = 10,260 seconds
Using the formula for exponential growth:
N = N₀ * e^(kt)
where N is the final number of bacteria, N₀ is the initial number of bacteria, e is the base of the natural logarithm (approximately 2.718), k is the rate constant, and t is the time interval.
Substituting the known values:
N = 100 * e^(0.0365 * 10,260)
Using a calculator, we can find the approximate number of bacteria:
N ≈ 6.927 * 10^56
So, approximately 6.927 * 10^56 bacteria will be present after 171 minutes.