Treating air as an ideal gas of diatomic molecules, calculate how much heat is required to raise the temperature of the air in an 7.47 m by 14.9 m by 2.87 m room from 19.9°C to 21.7°C at 101 kPa. Neglect the change in the number of moles of air in the room
The volume of the room is
7.47x14.9x2.87 = 319.4 m^3
= 319.4*10^6 liters.
If you assume the number of moles in the room remains the same, you are talking about a constant-volume heating. The specific heat for that, for a diatomic molecule, is (5/2)*R = 4.967 cal/mole C
The number of liters per mole at 19.9 C and 101 kPa (1 atm) is
22.4*(293.1/273.3) = 24.0 l/mol
Number of moles = 319.4*10^6 l/24.0 l/mol = 13.3*10^6 mol
heat required to raise temperature 1.8 C
= 1.8 C*4.967 cal/mole C*13.3*10^6 mole = ___
The volume of the room is
7.47x14.9x2.87 = 319.4 m^3
= 319.4*10^3 liters.
not 319.4*10^6 liters.
To calculate the amount of heat required to raise the temperature of the air in the room, we can use the formula:
Q = m * c * ΔT
Where:
Q = Amount of heat (in joules)
m = Mass of the air (in kilograms)
c = Specific heat capacity of air (in J/kg·K)
ΔT = Change in temperature (in Kelvin)
To calculate the mass of the air, we need to know the density of air. The ideal gas law can be used to find the density of air:
P = ρ * R * T
Where:
P = Pressure (in Pascals)
ρ = Density of air (in kilograms per cubic meter)
R = Gas constant (in J/(kg·K))
T = Temperature (in Kelvin)
We can rearrange this equation to solve for ρ:
ρ = P / (R * T)
Now we can calculate the density of air using the given temperature and pressure.
Next, to find the mass of the air in the room, we multiply the density by the volume of the room:
m = ρ * V
Once we have the mass, we can calculate the amount of heat required using the specific heat capacity of air. The specific heat capacity of air at constant pressure is approximately 1005 J/(kg·K).
Now let's calculate step by step:
Step 1: Convert the given temperature to Kelvin.
T1 = 19.9°C + 273.15 = 293.05 K
T2 = 21.7°C + 273.15 = 294.85 K
Step 2: Convert the given pressure to Pascals.
P = 101 kPa = 101,000 Pa
Step 3: Calculate the density of air.
R = 287.1 J/(kg·K) (gas constant for air)
ρ = P / (R * T1)
Step 4: Calculate the mass of air in the room.
V = 7.47 m * 14.9 m * 2.87 m = volume of the room
m = ρ * V
Step 5: Calculate the change in temperature.
ΔT = T2 - T1
Step 6: Calculate the amount of heat required.
Q = m * c * ΔT
Plug in the values and calculate Q.