To determine the increase in total energy of the molecules in the air when the temperature is raised from 20°C to 25°C, we can use the equation:
ΔE = (3/2) * N * k * ΔT
where: ΔE is the change in total energy, N is the total number of molecules in the room, k is the Boltzmann constant (approximately 1.38 x 10^-23 J/K), and ΔT is the change in temperature in Kelvin.
First, we need to convert the temperatures from Celsius to Kelvin:
Initial temperature (T1) = 20°C + 273.15 = 293.15 K Final temperature (T2) = 25°C + 273.15 = 298.15 K
Next, we calculate the change in temperature:
ΔT = T2 - T1 = 298.15 K - 293.15 K = 5 K
Assuming we have an ideal gas, we can estimate the total number of air molecules (N) in the room using the ideal gas law:
PV = NkT
where P is the pressure and V is the volume of the room. Let's assume a standard pressure of 1 atmosphere (atm).
For the sake of simplicity, let's say the room has a volume of 100 cubic meters. We can now calculate N:
N = PV / kT
N = (1 atm * 100 m^3) / (1.38 x 10^-23 J/K * 293.15 K)
N ≈ 2.34 x 10^26 molecules
Now we can calculate the change in total energy (ΔE):
ΔE = (3/2) * N * k * ΔT
ΔE = (3/2) * (2.34 x 10^26) * (1.38 x 10^-23 J/K) * 5 K
ΔE ≈ 2.4 x 10^4 Joules
Therefore, by raising the temperature from 20°C to 25°C in a room with the given assumptions, the total energy of all the molecules of air in the room increases by approximately 2.4 x 10^4 Joules.