A person takes in a breath of 13*C of air and holds it until it warms to 36.5*C. The air has an initial volume of 0.24 L and a mass of 0.00077kg.
a. Determine the work done by the air on the lungs if the pressure remains constant at 1 atm.
Answer in units of J.
b. Find the change in the internal energy of the air. Treat the air as a monatomic gas.
Answer in units of J.
c. Calculate the energy added to the air by heat.
Answer in units of J.
To solve this problem, we can use the ideal gas law and the first law of thermodynamics (also known as the energy conservation principle).
a. To determine the work done by the air on the lungs, we can use the equation:
Work = P * ΔV, where P is the pressure and ΔV is the change in volume.
Since the pressure remains constant at 1 atm, we can substitute P = 1 atm = 101325 Pa.
To find ΔV, we need to calculate the change in volume of the air. The volume can be obtained using the equation:
V = m / ρ, where V is the volume, m is the mass, and ρ is the density.
Since we know the initial volume is 0.24 L and the initial mass is 0.00077 kg, we can calculate the initial density:
ρ_initial = m / V_initial = 0.00077 kg / 0.24 L
Next, we need to calculate the final density using the final volume and mass. We know that volume is directly proportional to temperature, assuming constant pressure and amount of gas. Therefore, we can use the equation:
V_final / V_initial = T_final / T_initial
V_final = T_final * V_initial / T_initial
The final mass is the same as the initial mass, so we don't need to calculate it.
Now that we have the final density and initial density, we can find the change in volume:
ΔV = (m / ρ_final) - (m / ρ_initial)
Once we have calculated the change in volume, we can determine the work done:
Work = P * ΔV
b. The change in internal energy of the air can be calculated using the equation:
ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added to the system, and W is the work done by the system.
Since the problem states that the pressure remains constant, we can assume that the heat added to the air is equal to the change in internal energy. Therefore:
ΔU = Q
c. To calculate the energy added to the air by heat, we can use the equation:
Q = m * c * ΔT, where Q is the heat added, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Given that the air is treated as a monatomic gas, we can use the equation for the specific heat capacity of a monatomic ideal gas:
c = (3/2) * R, where R is the ideal gas constant.
Using the given initial and final temperatures, we can calculate the change in temperature:
ΔT = T_final - T_initial
Then we can plug all the values into the equation to find the energy added to the air by heat.