The temperature of a gas is -50° C. To what temperature the gas should be heated so that the rms speed is increased by 3 times?
- 223 K
- 669°C
- 3295°C
- 3097 K
The Correct Option is C
Approach Solution - 1
We know that,
\(V_{rms}\propto \sqrt{T}\)
\(\frac{V_{1}}{V_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}\)
\(⇒\) let the initial speed be \(v\).
As the speed increases by 3 times,
So, the final speed \(= 4v\)
\(⇒ \frac{v}{4v}\)
\(= \sqrt{\frac{223}{T}}\)
\(T=3568\text{ K}\)
So, the temperature in \(\degree{C} = 3568 - 273 = 3295\degree{C}\)
The correct answer is (C): \(3295°C\).
Approach Solution -2
The new velocity (V’rms) has been increased by 3 times the old velocity (Vrms).
Therefore, \(V’_{rms} = V_{rms} + 3 V_{rms} = 4V_{rms}\)
And, we also know Vrms is directly proportional to \(\sqrt{T}\)
Given, the initial temperature (T) = -50°C
Converting the temperature (T) to Kelvin: -50°C + 273 = 223K
Therefore initial temperature (T) in kelvin = 223K
If the speed increases to 4 times, the temperature should also increase by 16 times.
\(\frac{V’_{rms}}{V_{rms}} = \sqrt{\frac{T'}{T}}\)
\(\frac{4V_{rms}}{V_{rms}} = \sqrt \frac{T'}{223}\)
Therefore, Final Temperature (T’) \(= 16 \times 223 = 3568 K\)
Hence, Final Temperature (T’) in \(\degree C = 3568-273 = 3295 \degree C\)
So, the correct option is (C): \(3295 \degree C\)
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Concepts Used:
Kinetic Theory of Gases & Formulae
Kinetic theory of ideal gases is based on the molecular picture of matter. An ideal gas is a gas that follows Boyle's law, Charles' law, Gay Lussac's law, and Avogadro’s law.
- Gases have no shape or size and can be contained in vessels of any shape and size.
- Since the molecule of the gases is apart from each other, they have a negligible force of molecular interaction.
- Therefore, gases expand indefinitely and uniformly to fill the available space.
- Many scientists like Boyle and Newton tried to explain the behavior of gases, but Maxwell and Boltzmann developed the real theory in the 19th century.
- This theory is known as the Kinetic theory, which explains the behavior of gases.
The kinetic Theory of Gases is a classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The kinetic theory of gases describes a gas as a large number of identical submicroscopic particles, all of which are in constant, random, rapid motion.
Also Read: Kinetic Theory of Gases - Assumptions
Kinetic Theory of Gas Formulae:
Boltzmann’s Constant
kB = nR/N
kB is the Boltzmann’s constant
R is the gas constant
n is the number of moles
N is the number of particles in one mole (the Avogadro number)
Total Translational K.E of Gas
K.E = (3/2)nRT
n is the number of moles
R is the universal gas constant
T is the absolute temperature
Maxwell Distribution Law
Vrms ＞ V＞ Vp
Vrms is the RMS speed.
V is the Average speed.
Vp is the most probable speed.
RMS Speed (Vrms)
Vrms = \(\sqrt{8kt/m}\) =\(\sqrt{3RT/M}\)
R is the universal gas constant.
T is the absolute temperature.
M is the molar mass.
Average Speed
\(\overrightarrow{v} = \sqrt{8kt/πm} = \sqrt{8RT/πM}\)
Most Probable Speed (Vp)
\(V_ρ = \sqrt{2kt/m} = \sqrt{2RT/M}\)
The Pressure of Ideal Gas
\(P=\frac{1}{3}V^2rms\)
P is the density of molecules.
Equipartition of Energy
\(K=\frac{1}{2}K_BT\) for each degree of freedom.
K = (f/2) KвT for molecules having f degrees of freedom.
KB is the Boltzmann’s constant.
T is the temperature of the gas.
Internal Energy
U = (f/2) nRT
For n moles of an ideal gas.
Read About: Kinetic Theory of Gases Formulae