Question

В сосуде находится некоторое количество азота при давление 10 атм. Кинетическая энергия (поступательного и вращательного движений) всех молекул газа равна 581 Дж, их средняя квадратичная скорость 500 м/с. Найти массу и температуру азота, а также объем сосуда?

Answer 1

Азот - двухатомный газ, поэтому сумму поступательных и вращательных энергий его движения нужно записывать в виде

$E = \frac{5}{2}\nu R T$

При этом доля энергии, связанная с поступательным движением выражается с одной стороны через ср. квад. скорость, а с другой через температуру

$E_k = \frac{3}{2}\nu R T = \frac{\nu Mv^2}{2}\\\\ T = \frac{Mv^2}{3R} = \frac{0.028\cdot500^2}{21.93} \approx 281 K$

$3 R T = Mv^2}\\\\ M/RT = 3/v^2\\\\ E = \frac{5}{2}\frac{m}{M} RT\\\\ m = \frac{2ME}{5RT} = \frac{6E}{5v^2} \approx 2.8 \cdot 10^{-3}$

2.8 грамма или около 0.1 моль

Объем

$V = \frac{\nu R T}{p} = \frac{0.1 \cdot 8.31\cdot 281}{10^6} = 233\cdot10^{-6}$

Или 233 см. куб

Answer 2

Calculation of Mass, Temperature, and Volume of Nitrogen Gas

To find the mass, temperature, and volume of the nitrogen gas in the vessel, we can use the following equations:

1. The kinetic energy of the gas molecules is given by the equation:

E = (3/2) * k * T

where E is the total kinetic energy, k is the Boltzmann constant, and T is the temperature.

2. The average kinetic energy of the gas molecules is related to the average square velocity by the equation:

E = (1/2) * m * v^2

where m is the mass of a gas molecule and v is the average square velocity.

3. The ideal gas law equation relates the pressure, volume, and temperature of a gas:

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.

Let's use these equations to find the mass, temperature, and volume of the nitrogen gas.

Given: - Pressure (P) = 10 atm - Total kinetic energy (E) = 581 J - Average square velocity (v) = 500 m/s

First, we need to find the mass of the nitrogen gas molecule (m). We can rearrange equation 2 to solve for m:

m = (2 * E) / (v^2)

Substituting the given values:

m = (2 * 581 J) / (500^2 m^2/s^2)

Calculating the mass using the given values:

m ≈ 0.00232 kg

Next, we can find the temperature (T) using equation 1. Rearranging the equation, we have:

T = (2 * E) / (3 * k)

Substituting the given values:

T = (2 * 581 J) / (3 * k)

To find the value of the Boltzmann constant (k), we can refer to a reliable source. According to the value of the Boltzmann constant is approximately 1.38 * 10^-23 J/K.

Substituting the value of k:

T ≈ (2 * 581 J) / (3 * 1.38 * 10^-23 J/K)

Calculating the temperature using the given values:

T ≈ 2.81 * 10^22 K

Finally, we can find the volume (V) of the vessel using the ideal gas law equation. Rearranging the equation, we have:

V = (n * R * T) / P

Since we don't have the number of moles (n), we can use the molar mass of nitrogen to calculate it. According to the molar mass of nitrogen is approximately 28 g/mol.

Converting the mass of the nitrogen gas molecule to moles:

n = (0.00232 kg) / (0.028 kg/mol)

Calculating the number of moles:

n ≈ 0.083 moles

Substituting the values of n, R (the ideal gas constant), and T into the equation:

V = (0.083 mol * 8.314 J/(mol*K) * 2.81 * 10^22 K) / (10 atm)

Converting atm to Pa (1 atm = 101325 Pa):

V = (0.083 mol * 8.314 J/(mol*K) * 2.81 * 10^22 K) / (10 * 101325 Pa)

Calculating the volume using the given values:

V ≈ 0.0022 m^3

Therefore, the mass of the nitrogen gas is approximately 0.00232 kg, the temperature is approximately 2.81 * 10^22 K, and the volume of the vessel is approximately 0.0022 m^3.

Please note that the calculated temperature seems unusually high. It is possible that there may be an error in the given values or calculations.