Question

Плотность газа в баллоне электрической лампы ρ=0,9 кг/м3. При горении лампы давление в ней возросло с p1=8*104 Па до p2=1,1*105 Па. На сколько увеличилась при этом средняя скорость молекул газа?

Answer 1

Calculation of the Increase in Average Gas Molecule Speed

To calculate the increase in the average speed of gas molecules when the pressure inside the lamp increases from p1 = 8*10^4 Pa to p2 = 1.1*10^5 Pa, we can use the ideal gas law and the relationship between pressure and average molecular speed.

The ideal gas law states that PV = nRT, where: - P is the pressure of the gas, - V is the volume of the gas, - n is the number of moles of gas, - R is the ideal gas constant, and - T is the temperature of the gas in Kelvin.

Since the volume and the number of moles of gas remain constant, we can rewrite the ideal gas law as P/T = constant.

According to the kinetic theory of gases, the average speed of gas molecules is directly proportional to the square root of the temperature. Therefore, we can write v1/v2 = sqrt(T1/T2), where: - v1 is the initial average speed of gas molecules, - v2 is the final average speed of gas molecules, - T1 is the initial temperature of the gas, and - T2 is the final temperature of the gas.

Since the pressure and temperature are directly proportional, we can write P1/T1 = P2/T2.

Now, let's solve for the increase in the average speed of gas molecules.

Solution:

Given: - Initial pressure, p1 = 8*10^4 Pa - Final pressure, p2 = 1.1*10^5 Pa - Initial density of the gas, ρ = 0.9 kg/m^3

To find the increase in the average speed of gas molecules, we need to calculate the initial and final temperatures of the gas.

Using the ideal gas law, we can write p1 = ρRT1 and p2 = ρRT2.

Simplifying the equations, we have T1 = p1 / (ρR) and T2 = p2 / (ρR).

Substituting the given values, we get: - T1 = (8*10^4 Pa) / (0.9 kg/m^3 * R) - T2 = (1.1*10^5 Pa) / (0.9 kg/m^3 * R)

Now, let's calculate the initial and final temperatures.

Using the value of the ideal gas constant, R = 8.314 J/(mol·K), we can substitute it into the equations to find the temperatures.

Calculating the temperatures: - T1 = (8*10^4 Pa) / (0.9 kg/m^3 * 8.314 J/(mol·K)) - T2 = (1.1*10^5 Pa) / (0.9 kg/m^3 * 8.314 J/(mol·K))

Now, let's calculate the increase in the average speed of gas molecules using the relationship v1/v2 = sqrt(T1/T2).

Substituting the calculated temperatures into the equation, we get: - v1/v2 = sqrt((8*10^4 Pa) / (1.1*10^5 Pa))

Calculating the increase in the average speed of gas molecules: - v1/v2 = sqrt(0.727)

Therefore, the increase in the average speed of gas molecules is approximately 0.853.

Please note that the above calculation assumes an ideal gas behavior and does not take into account any other factors that may affect the speed of gas molecules.