On a cold day, you take a breath, inhaling 0.50 L of air whose initial temperature
is −10°C. In your lungs, its temperature is raised to 37°C. Assume that the
pressure is 101 kPa and that the air may be treated as an ideal gas. What is the
total change in translational kinetic energy of the air you inhaled?
To find the total change in translational kinetic energy, we first need to know the final volume of the air after it has been warmed to 37°C in the lungs. We can use the ideal gas law for this:
PV = nRT
Where P is pressure, V is volume, n is the number of moles of the gas, R is the ideal gas constant, and T is the temperature in Kelvins. The ideal gas constant R has a value of 8.314 J/mol·K. Note that the gas law applies to both the initial and final states of the air, so we can write two equations:
P * V_initial = n * R * T_initial
P * V_final = n * R * T_final
Dividing the second equation by the first one, we get:
V_final / V_initial = T_final / T_initial
We are given V_initial = 0.50 L and T_initial = -10°C, which we need to convert to Kelvin: T_initial_K = -10 + 273.15 = 263.15 K. The final temperature is 37°C, which is equivalent to T_final_K = 37 + 273.15 = 310.15 K. Plugging in these values:
V_final / 0.50 L = 310.15 K / 263.15 K
Solving for V_final:
V_final = 0.50 L * (310.15 K / 263.15 K) = 0.589 L
Now we can find the change in translational kinetic energy. The formula for the (average) translational kinetic energy per molecule of an ideal gas is:
K.E. = (3/2) * k * T
Where k is the Boltzmann constant (1.38 * 10^(-23) J/K). The total translational kinetic energy of the gas is the product of the kinetic energy per molecule and the total number of molecules (N):
Total K.E. = N * (3/2) * k * T
We can relate the number of molecules (N) to the number of moles (n) through the Avogadro constant (NA = 6.022 * 10^(23) molecules/mol)
N = n * NA
Now we can write the total kinetic energy for the initial and final states:
Total K.E._initial = n * NA * (3/2) * k * T_initial_K
Total K.E._final = n * NA * (3/2) * k * T_final_K
The change in kinetic energy is the difference between the final and initial values:
ΔK.E. = Total K.E._final - Total K.E._initial
Factoring out the common terms (n * NA * (3/2) * k), we get:
ΔK.E. = n * NA * (3/2) * k * (T_final_K - T_initial_K)
In order to determine n, we need to rearrange the ideal gas law for the initial state:
n = (P * V_initial) / (R * T_initial_K)
Plugging in the known values, we have:
n = (101 kPa * 0.50 L) / (8.314 J/mol·K * 263.15 K)
Note that 1 kPa = 1000 Pa = 1000 J/m³, and 1 L = 0.001 m³. With this conversion, we have:
n = (101 * 1000 J/m³ * 0.50 * 0.001 m³) / (8.314 J/mol·K * 263.15 K)
n ≈ 0.0232 mol
Now, we can plug in the values for the change in kinetic energy:
ΔK.E. = 0.0232 mol * 6.022 * 10^(23) molecules/mol * (3/2) * 1.38 * 10^(-23) J/K * (310.15 K - 263.15 K)
ΔK.E. ≈ 1856.8 J
The total change in translational kinetic energy of the air you inhaled is approximately 1857 J.
To calculate the total change in translational kinetic energy of the inhaled air, we can use the equation:
ΔKE = 3/2 * R * ΔT
Where:
ΔKE is the change in translational kinetic energy
R is the gas constant
ΔT is the change in temperature
First, we need to convert the temperatures to Kelvin. The temperature in the lungs is 37°C + 273.15 = 310.15 K, and the initial temperature is -10°C + 273.15 = 263.15 K.
Next, we need to calculate the change in temperature:
ΔT = 310.15 K - 263.15 K = 47 K
Now we can calculate the change in translational kinetic energy:
ΔKE = 3/2 * R * ΔT
The gas constant, R, is approximately 8.314 J/(mol·K).
Substituting the values, we have:
ΔKE = 3/2 * 8.314 J/(mol·K) * 47 K
Simplifying the equation:
ΔKE = 3 * 8.314 J/(mol·K) * 47 K / 2
Calculating the value:
ΔKE = 586.305 J
Therefore, the total change in translational kinetic energy of the air you inhaled is approximately 586.305 J.
To find the total change in translational kinetic energy of the air you inhaled, we need to calculate the initial and final kinetic energies and then find the difference.
Translational kinetic energy is given by the formula:
KE = (3/2) * n * R * T
where KE is the kinetic energy, n is the number of moles of gas, R is the ideal gas constant (8.314 J/(mol·K)), and T is the temperature in Kelvin.
First, let's convert the temperatures to Kelvin. The initial temperature is -10°C, so in Kelvin, it would be:
T_initial = -10 + 273.15 = 263.15 K
The final temperature is 37°C, so in Kelvin, it would be:
T_final = 37 + 273.15 = 310.15 K
Now, we need to find the change in kinetic energy.
Step 1: Calculate the initial kinetic energy.
For the initial state, we know the temperature and the volume of air inhaled. However, we do not have information about the number of moles of the gas. To calculate the number of moles, we can use the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
From the question, we are given the pressure (101 kPa) and the volume (0.50 L). Substituting these values into the ideal gas law and solving for n:
n_initial = PV / (RT_initial)
Step 2: Calculate the final kinetic energy.
Using the same process, we can calculate the number of moles for the final state:
n_final = PV / (RT_final)
Step 3: Calculate the initial and final kinetic energy.
Using the formula KE = (3/2) * n * R * T, we can calculate the initial and final kinetic energies:
KE_initial = (3/2) * n_initial * R * T_initial
KE_final = (3/2) * n_final * R * T_final
Step 4: Find the total change in kinetic energy.
The total change in kinetic energy is the difference between the initial and final kinetic energies:
ΔKE = KE_final - KE_initial
Now, you can plug in the calculated values to find the total change in translational kinetic energy of the air you inhaled.