Respuesta :
Answer: 9 times of the initial temperature must it be raised to triple the rms speed of its molecules.
Explanation:
[tex]\mu_{rms}=\sqrt[2]{\frac{3RT}{M}}[/tex] (Root Mean Square of the molecules)
[tex]\mu_{rms}[/tex] = Root mean square speed of the molecules
R = Universal gas constant=[tex]8.314 J/ mol K[/tex]
T= Temperature in Kelvins
M = Molar mass in kg/mol
Let the initial root mean square speed of its molecules be [tex]\mu_{rms}[/tex] at temperature T.
Let the final root mean square speed of its molecules be [tex]\mu_{rms}'=3\times \mu_{rms}[/tex] at temperature T'.
[tex]\mu_{rms}'=\sqrt[2]{\frac{3RT'}{M}}=3\sqrt[2]{\frac{3RT}{M}}[/tex]
[tex]T'=9\times T[/tex]
9 times of the initial temperature must it be raised to triple the rms speed of its molecules.
The new temperature must be raised to [tex]\boxed{{\text{9 times}}}[/tex] the original temperature in order to triple the rms speed of the molecules.
Further Explanation:
Gases neither have a definite shape nor a definite volume. These are highly compressible and have an irregular or disordered arrangement of its constituent particles. The intermolecular forces in the gases are the weakest and thus the motion of particles in a gas is very high. Oxygen, hydrogen and carbon dioxide are examples of gases.
The kinetic theory is based on the following postulates:
1. Gas molecules have a large collection of individual particles with empty space between them and the volume of each particle is very small as compared to the volume of the whole gas.
2. The gas particles are in straight-line motion or random motion until they are not collided with the wall of the container or with each other.
3. The collision between the gas particles and the wall of the containers is an elastic collision that means molecules exchange energy but they don’t lose any energy during the collision. So the total kinetic energy is constant.
The formula to calculate the root mean square speed of the gas is as follows:
[tex]{\mu _{{\text{rms}}}}=\sqrt{\frac{{{\text{3RT}}}}{{\text{M}}}}[/tex] …… (1)
Here,
[tex]{\mu _{{\text{rms}}}}[/tex] is the root mean square speed of gas.
R is the universal gas constant.
T is the absolute temperature.
M is the molar mass of gas.
It is given that rms speed is tripled. Consider the new rms speed to be [tex]\mu {'_{{\text{rms}}}}[/tex] . The formula to calculate [tex]\mu {'_{{\text{rms}}}}[/tex] is as follows:
[tex]\mu {'_{{\text{rms}}}}=\sqrt{\frac{{{\text{3RT'}}}}{{\text{M}}}}[/tex] …… (2)
Here T’ is the new temperature when rms speed is tripled.
As per the given conditions, the new rms speed is 3 times the original rms speed. This is expressed using equation (3).
[tex]\mu {'_{{\text{rms}}}}=3{\mu _{{\text{rms}}}}[/tex] …… (3)
Substitute the values of [tex]{\mu _{{\text{rms}}}}[/tex] and [tex]\mu {'_{{\text{rms}}}}[/tex] from equations (1) and (2) in equation (3).
[tex]\sqrt{\frac{{{\text{3RT'}}}}{{\text{M}}}}=3\left({\sqrt{\frac{{{\text{3RT}}}}{{\text{M}}}}} \right)[/tex] …… (4)
All the quantities, except for T and T’ are the same on both sides of equation (4). So this equation becomes,
[tex]{\text{T}}'=9{\text{T}}[/tex]
Therefore the new temperature must be raised to 9 times the original temperature in order to triple the rms speed of the molecules.
Learn more:
1. Which will have the greatest average speed at 355 K?: https://brainly.com/question/1852048
2. Which statement is true for Boyle’s law: https://brainly.com/question/1158880
Answer details:
Grade: High School
Subject: Chemistry
Chapter: Ideal gas equation
Keywords: rms, absolute temperature, gas, universal gas constant, R, T, M, molar mass, T’, 9 times, tripled.
