Inside a sphere with a diameter of 20 cm there is oxygen, while

Inside a sphere with a diameter of 20 cm there is oxygen at a temperature of 17 °C. It is necessary to determine the gas pressure and the number of molecules in 1 cm³, if the free path of the molecules is equal to the diameter of the vessel (the molecules do not experience collisions with each other).

Hopefully:

• Sphere diameter: d = 20 cm = 0.2 m
• Oxygen temperature: T = 17 °C = 290 K
• Molecular mean free path: l = d = 0.2 m

Required quantities:

• Gas pressure: p
• Number of molecules in 1 cm³: n

Answer:

Gas pressure can be found using the formula:

p = n m v² / 3, where

• n - number of molecules per unit volume
• m - mass of one oxygen molecule
• v - average speed of molecules

Mass of one oxygen molecule:

m = M / N_A, Where

• M - molar mass of oxygen
• N_A - Avogadro's constant

The molar mass of oxygen is equal to:

M = 32 g/mol

Avogadro's constant is:

N_A = 6,022 × 10²³ mole^-1

Then:

m = 32 × 10^-3 kg / 6,022 × 10²³ mole^-1 ≈ 5,31 × 10^-26 kg

The average speed of molecules can be found using the formula:

v = (8 × k T / (Pi m))^1/2, Where

• k - Boltzmann constant

Boltzmann's constant is:

k = 1,38 × 10^-23 J/C

Then:

v = (8 × 1,38 × 10^-23 J/K × 290 K / (Pi × 5,31 × 10^-26 kg))^1/2 ≈ 468 m/s

The number of molecules per unit volume can be found using the formula:

n = N / V, Where

• N - total number of molecules in the sphere
• V - volume of the sphere

The total number of molecules in a sphere can be found using the formula:

N = N_Am / M × V, Where

• N_A - Avogadro's constant

Then:

N = 6,022 × 10²³ mole^-1 × 0.032 kg/mol / 0.2 m ≈ 4.83 × 10²¹ molecules

Sphere volume:

V = 4/3 × Pi (d/2)³ = 4/3 × 3.14 × (0.1 m)³ ≈ 0.0042 m³

Then:

n = 4,83 × 10²¹ molecule / 0.0042 m³ ≈ 1,15 × 10²⁵ molecule/m³

So, we have determined the gas pressure and the number of molecules per 1 cm³:

p = n m v² / 3 ≈ 5,7 × 10² Pa

n ≈ 1,15 × 10²⁵ molecule/m³

Answer: gas pressure is approximately 570 Pa, the number of molecules per 1 cm³ approximately equal to 1.15 × 10²⁵.

Task solved.

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The problem is formulated as follows: “Inside a sphere with a diameter of 20 cm there is oxygen at a temperature of 17 ° C. Determine the gas pressure and the number of molecules in 1 cm³, if the free path of the molecules is equal to the diameter of the vessel (the molecules do not experience collisions with each other)."

In this product we provide a brief description of the conditions of the problem, formulas and laws used in the solution, the derivation of the calculation formula and the answer. All this is presented in a beautiful html code, which allows you to quickly and conveniently familiarize yourself with the material.

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I present to you a beautifully designed digital product - a solution to a physics problem! In this product you will find a detailed and understandable solution to problem 20603, which may arise for students and schoolchildren when studying thermodynamics and the kinetic theory of gases.

The problem is formulated as follows: “Inside a sphere with a diameter of 20 cm there is oxygen at a temperature of 17 ° C. Determine the gas pressure and the number of molecules in 1 cm^3 if the free path of the molecules is equal to the diameter of the vessel (the molecules do not experience collisions with each other).”

To solve the problem we use the following formulas:

1. Gas pressure can be found using the formula: p = n m v^2/3, where n is the number of molecules per unit volume, m is the mass of one oxygen molecule, v is the average speed of the molecules.

2. The mass of one oxygen molecule can be found using the formula: m = M/NA, where M is the molar mass of oxygen, NA is Avogadro’s constant.

3. The average speed of molecules can be found using the formula: v = (8 k T/π m)^1/2, where k is Boltzmann’s constant, T is the temperature of oxygen.

4. The number of molecules per unit volume can be found using the formula: n = N/V, where N is the total number of molecules in the sphere, V is the volume of the sphere.

5. The total number of molecules in the sphere can be found using the formula: N = NA m/M × V.

Using the data from the problem conditions and formulas, we obtain the following results:

Sphere diameter: d = 20 cm = 0.2 m Oxygen temperature: T = 17 °C = 290 K Molecular mean free path: l = d = 0.2 m

Molar mass of oxygen: M = 32 g/mol Avogadro's constant: NA = 6.022 × 10^23 mol^-1

Mass of one oxygen molecule: m = M/NA ≈ 5.31 × 10^-26 kg Average molecular speed: v ≈ 468 m/s Sphere volume: V ≈ 0.0042 m^3 Total number of molecules in the sphere: N ≈ 4.83 × 10^21 molecules Number of molecules per unit volume: n ≈ 1.15 × 10^25 molecules/m^3 Gas pressure: p ≈ 570 Pa

Answer: the gas pressure is approximately 570 Pa, the number of molecules in 1 cm^3 is approximately 1.15 × 10^25.

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inside a sphere with a diameter of 20 cm there is oxygen at a temperature of 17 °C. For this system, it is necessary to determine the gas pressure and the number of molecules in 1 cm^3, provided that the free path of the molecules is equal to the diameter of the vessel. This problem uses the laws of gas physics, namely the equation of state of an ideal gas and the formula for calculating the number of molecules per unit volume.

Answer:

To solve the problem, it is necessary to use the equation of state of an ideal gas: PV = nRT, where P is gas pressure, V is volume, n is the amount of substance, R is the universal gas constant, T is temperature.

A formula is also used to calculate the number of molecules per unit volume: N/V = n/Na, where N is the number of molecules, Na is Avogadro’s number.

First you need to determine the amount of substance n. To do this, it is necessary to express n from the ideal gas equation of state: n = PV/RT.

We substitute known values: P - unknown V = (4/3)πr^3 = (4/3)π(0.1 m)^3 = 4.19×10^-4 m^3, R = 8.31 J/(mol*K), T = 17 + 273 = 290 K.

Thus, we get: n = PV/(RT).

Next, it is necessary to determine the gas pressure P. To do this, we will use the condition of the problem that the free path of the molecules is equal to the diameter of the vessel. The free path of molecules is determined by the formula λ = kT/(√2πd^2p), where k is Boltzmann’s constant, d is the diameter of the molecules, p is the gas pressure.

Thus, d = 2r = 0.2 m, k = 1.38×10^-23 J/K, λ = d. We substitute known values:

d = 0.2 m, k = 1.38×10^-23 J/K, T = 17 + 273 = 290 K.

Then we get:

d = kT/(√2πd^2p)

p = kT/(√2πd^3).

We substitute known values:

k = 1.38×10^-23 J/K, T = 290 K, d = 0.2 m.

Thus we get:

p = 1.38×10^-23 * 290 / (√2π * (0.2)^3) = 0.0262 Pa.

Next, it is necessary to determine the number of molecules per unit volume. To do this, we use the formula N/V = n/Na.

We substitute known values:

n = PV/(RT) = 0.02624.19×10^-4/(8.31290) = 1.16×10^-7 mol, Na = 6.02×10^23 mol^-1, V = (4/3)πr^3 = 4.19×10^-4 m^3.

Thus we get:

N/V = n/Na = 1.16×10^-7 / 6.02×10^23 * 10^6= 1.93×10^12 м^-3.

Thus, in a given sphere with a diameter of 20 cm at a temperature of 17 °C there is oxygen at a pressure of 0.0262 Pa and the number of molecules per 1 cm^3 is 1.93 × 10^12.

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