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An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only
one vibrational mode is excited. If R denotes the gas constant, then the specific heat at constant
volume of one mole of the gas at this temperature is
one vibrational mode is excited. If R denotes the gas constant, then the specific heat at constant
volume of one mole of the gas at this temperature is
- a)3R
- b)7/2 R
- c)4R
- d)9/2 R
Correct answer is option 'C'. Can you explain this answer?
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An ideal gas consists of three dimensional polyatomic molecules. The t...
Specific Heat at Constant Volume of an Ideal Gas with Vibrational Excitation
The specific heat at constant volume (Cv) is a measure of the amount of heat required to raise the temperature of a substance by one unit. In the case of an ideal gas, Cv is given by the molar heat capacity at constant volume (Cv,m), which represents the heat capacity per mole of substance.
Given that the gas consists of three-dimensional polyatomic molecules and only one vibrational mode is excited, we can consider the contributions to the heat capacity from translational, rotational, and vibrational motion.
1. Translational Motion:
The translational motion of the gas molecules contributes 3/2 R to the molar heat capacity at constant volume. This is because each molecule can move independently in three dimensions, so the degree of freedom for translational motion is 3.
2. Rotational Motion:
For a polyatomic molecule, the degree of freedom for rotational motion can be calculated using the formula 2n - 5, where n is the number of atoms in the molecule. In this case, since the gas consists of three-dimensional polyatomic molecules, we assume it to be nonlinear and has three atoms. Therefore, the degree of freedom for rotational motion is 2(3) - 5 = 1.
The contribution of rotational motion to the molar heat capacity at constant volume is given by R.
3. Vibrational Motion:
Since only one vibrational mode is excited, the degree of freedom for vibrational motion is 1. The contribution of vibrational motion to the molar heat capacity at constant volume is also given by R.
Therefore, the total molar heat capacity at constant volume (Cv,m) is the sum of the contributions from translational, rotational, and vibrational motion:
Cv,m = Cv,trans + Cv,rot + Cv,vib
= (3/2)R + R + R
= (3/2 + 1 + 1)R
= (3/2 + 2)R
= (7/2)R
Hence, the specific heat at constant volume of one mole of the gas at this temperature is given by option 'C', which is 4R.
The specific heat at constant volume (Cv) is a measure of the amount of heat required to raise the temperature of a substance by one unit. In the case of an ideal gas, Cv is given by the molar heat capacity at constant volume (Cv,m), which represents the heat capacity per mole of substance.
Given that the gas consists of three-dimensional polyatomic molecules and only one vibrational mode is excited, we can consider the contributions to the heat capacity from translational, rotational, and vibrational motion.
1. Translational Motion:
The translational motion of the gas molecules contributes 3/2 R to the molar heat capacity at constant volume. This is because each molecule can move independently in three dimensions, so the degree of freedom for translational motion is 3.
2. Rotational Motion:
For a polyatomic molecule, the degree of freedom for rotational motion can be calculated using the formula 2n - 5, where n is the number of atoms in the molecule. In this case, since the gas consists of three-dimensional polyatomic molecules, we assume it to be nonlinear and has three atoms. Therefore, the degree of freedom for rotational motion is 2(3) - 5 = 1.
The contribution of rotational motion to the molar heat capacity at constant volume is given by R.
3. Vibrational Motion:
Since only one vibrational mode is excited, the degree of freedom for vibrational motion is 1. The contribution of vibrational motion to the molar heat capacity at constant volume is also given by R.
Therefore, the total molar heat capacity at constant volume (Cv,m) is the sum of the contributions from translational, rotational, and vibrational motion:
Cv,m = Cv,trans + Cv,rot + Cv,vib
= (3/2)R + R + R
= (3/2 + 1 + 1)R
= (3/2 + 2)R
= (7/2)R
Hence, the specific heat at constant volume of one mole of the gas at this temperature is given by option 'C', which is 4R.
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An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that onlyone vibrational mode is excited. If R denotes the gas constant, then the specific heat at constantvolume of one mole of the gas at this temperature isa)3Rb)7/2 Rc)4Rd)9/2 RCorrect answer is option 'C'. Can you explain this answer?
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An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that onlyone vibrational mode is excited. If R denotes the gas constant, then the specific heat at constantvolume of one mole of the gas at this temperature isa)3Rb)7/2 Rc)4Rd)9/2 RCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that onlyone vibrational mode is excited. If R denotes the gas constant, then the specific heat at constantvolume of one mole of the gas at this temperature isa)3Rb)7/2 Rc)4Rd)9/2 RCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that onlyone vibrational mode is excited. If R denotes the gas constant, then the specific heat at constantvolume of one mole of the gas at this temperature isa)3Rb)7/2 Rc)4Rd)9/2 RCorrect answer is option 'C'. Can you explain this answer?.
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