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If a gas has n degree of freedom, ratio of principal specific heats of the gas is
  • a)
    1+ 2/n
  • b)
    2n
  • c)
    1 - 2/n
  • d)
    -2n
Correct answer is option 'A'. Can you explain this answer?
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If a gas has n degree of freedom, ratio of principal specific heats of...
Let us consider 1 mole of an ideal gas at kelvin temperature T. It has N molecules (Avogadro's number). The internal energy of an ideal gas is entirely kinetic. The average KE per molecule of a ideal gas is ½ ​nkT (k is boltzman constant), where n is degree of freedom. Therefore, the internal energy of one mole of a gas would be
 
E=N(1/2​nKT)=1/2​nRT             (∵k=R/N​)
 
Now, Cv​=dE/dT​=n/2 ​R
 
and Cp​=n/2 ​R+R=(n/2​+1)R
 
​Cp/ Cv ​​=​(n/2​+1)R/n/2​=(1+2/n​)
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If a gas has n degree of freedom, ratio of principal specific heats of...
Here principle specific heats of gas are nothing but specific heats at constant pressure and volume.
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If a gas has n degree of freedom, ratio of principal specific heats of...
Principal specific heats of a gas are the specific heats at constant pressure (Cp) and constant volume (Cv). The ratio of these two specific heats is denoted by γ (gamma) and is given by the equation:

γ = Cp / Cv

To find the ratio of principal specific heats, we need to determine the number of degrees of freedom (n) for the gas.

Explanation:
1. Degrees of Freedom:
- In physics, degrees of freedom refer to the number of independent parameters or variables that define the state of a system.
- For a gas, the degrees of freedom represent the number of ways in which the gas molecules can move. It depends on the type of motion (translational, rotational, vibrational) allowed for the gas molecules.
- For a monatomic gas (e.g., noble gases like helium, neon, etc.), the only allowed motion is translational, so it has three degrees of freedom.
- For a diatomic gas (e.g., oxygen, nitrogen, etc.), both translational and rotational motion are allowed. In this case, the gas molecules have five degrees of freedom (three translational and two rotational).
- For a general gas, the total number of degrees of freedom (n) can be given by the equation:
n = 3 + f
where f represents the number of additional degrees of freedom due to vibrational or other possible motions.

2. Ratio of Principal Specific Heats:
- For an ideal gas, the specific heat at constant pressure (Cp) is related to the number of degrees of freedom (n) by the equation:
Cp = (n / 2) R
where R is the gas constant.
- Similarly, the specific heat at constant volume (Cv) is given by:
Cv = ((n / 2) - 1) R
- Substituting these values in the equation for γ, we get:
γ = Cp / Cv = [(n / 2) R] / [((n / 2) - 1) R]
Simplifying this equation, we find:
γ = (n / 2) / ((n / 2) - 1) = n / (n - 2)

3. Finding the Ratio for the Given Number of Degrees of Freedom:
- For the given gas with n degrees of freedom, the ratio of principal specific heats (γ) is:
γ = n / (n - 2)
- Since we are asked for the ratio in terms of n, we can rewrite this equation as:
γ = (n - 2 + 2) / (n - 2) = 1 + 2 / (n - 2)
- Comparing this equation with the given options, we can see that the correct answer is option 'A': 1 + 2/n.
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Importance of high specific heat capacity of water for life Specific heat capacity of a substance is the amount of heat required to raise the temperature of that substance by 1 K. It is expressed in the units J/ (kg K). A high specific heat of a substance means that a large amount of heat is required to raise the temperature of the substance. Water has the highest known specific heat capacity. Its specific heat capacity is 4.186 K J/ (kg K) i.e. to raise the temperature of 1 kg of water by 1 Kelvin it requires 4.186 KJ of heat. For comparison sake, Copper requires only 385 Joules of heat to raise 1 kilogram of copper by 1 Kelvin. It also interesting to know that the specific heat capacities in two other phases of water (i.e. ice and water vapour) are less than this. High specific heat of water is mainly due to the presence of a large number of hydrogen bonds between molecules of water. On a beach on a sunny day, it is noticed that the sand is often quite hot to walk on, but the water is always cool, even in the shallows. This is because sand has a lower specific heat capacity than that of water. Sand takes less energy to raise the temperature by one degree. Because water has a high specific heat capacity, it requires more energy to raise the temperature by one degree. Since the sun puts same rate of energy on water and sand, which heats up sand more quickly and water more slowly. Water covers around 70% of the Earth's surface and its high specific heat plays a very important role to sustain life in the earth. It is able to absorb a lot of heat without a significant rise in the temperature. When temperatures decrease, the heat which is stored is released, restraining a rapid drop in temperature. The combined effect is the buffering of heat. A relatively constant temperature without sudden rise and drop is essential to sustain life. Hence water is important for life.Q. To raise the temperature of 1 kg of water and 1 kg of copper by 1 Kelvin, 4.186 KJ and 385 Joule of heat are required respectively. Which one will get heated up faster if exposed to sun?

Importance of high specific heat capacity of water for life Specific heat capacity of a substance is the amount of heat required to raise the temperature of that substance by 1 K. It is expressed in the units J/ (kg K). A high specific heat of a substance means that a large amount of heat is required to raise the temperature of the substance. Water has the highest known specific heat capacity. Its specific heat capacity is 4.186 K J/ (kg K) i.e. to raise the temperature of 1 kg of water by 1 Kelvin it requires 4.186 KJ of heat. For comparison sake, Copper requires only 385 Joules of heat to raise 1 kilogram of copper by 1 Kelvin. It also interesting to know that the specific heat capacities in two other phases of water (i.e. ice and water vapour) are less than this. High specific heat of water is mainly due to the presence of a large number of hydrogen bonds between molecules of water. On a beach on a sunny day, it is noticed that the sand is often quite hot to walk on, but the water is always cool, even in the shallows. This is because sand has a lower specific heat capacity than that of water. Sand takes less energy to raise the temperature by one degree. Because water has a high specific heat capacity, it requires more energy to raise the temperature by one degree. Since the sun puts same rate of energy on water and sand, which heats up sand more quickly and water more slowly. Water covers around 70% of the Earth's surface and its high specific heat plays a very important role to sustain life in the earth. It is able to absorb a lot of heat without a significant rise in the temperature. When temperatures decrease, the heat which is stored is released, restraining a rapid drop in temperature. The combined effect is the buffering of heat. A relatively constant temperature without sudden rise and drop is essential to sustain life. Hence water is important for life.Q. Which statement is false?

Importance of high specific heat capacity of water for life Specific heat capacity of a substance is the amount of heat required to raise the temperature of that substance by 1 K. It is expressed in the units J/ (kg K). A high specific heat of a substance means that a large amount of heat is required to raise the temperature of the substance. Water has the highest known specific heat capacity. Its specific heat capacity is 4.186 K J/ (kg K) i.e. to raise the temperature of 1 kg of water by 1 Kelvin it requires 4.186 KJ of heat. For comparison sake, Copper requires only 385 Joules of heat to raise 1 kilogram of copper by 1 Kelvin. It also interesting to know that the specific heat capacities in two other phases of water (i.e. ice and water vapour) are less than this. High specific heat of water is mainly due to the presence of a large number of hydrogen bonds between molecules of water. On a beach on a sunny day, it is noticed that the sand is often quite hot to walk on, but the water is always cool, even in the shallows. This is because sand has a lower specific heat capacity than that of water. Sand takes less energy to raise the temperature by one degree. Because water has a high specific heat capacity, it requires more energy to raise the temperature by one degree. Since the sun puts same rate of energy on water and sand, which heats up sand more quickly and water more slowly. Water covers around 70% of the Earth's surface and its high specific heat plays a very important role to sustain life in the earth. It is able to absorb a lot of heat without a significant rise in the temperature. When temperatures decrease, the heat which is stored is released, restraining a rapid drop in temperature. The combined effect is the buffering of heat. A relatively constant temperature without sudden rise and drop is essential to sustain life. Hence water is important for life.Q. Specific heat capacity of water is

Importance of high specific heat capacity of water for life Specific heat capacity of a substance is the amount of heat required to raise the temperature of that substance by 1 K. It is expressed in the units J/ (kg K). A high specific heat of a substance means that a large amount of heat is required to raise the temperature of the substance. Water has the highest known specific heat capacity. Its specific heat capacity is 4.186 K J/ (kg K) i.e. to raise the temperature of 1 kg of water by 1 Kelvin it requires 4.186 KJ of heat. For comparison sake, Copper requires only 385 Joules of heat to raise 1 kilogram of copper by 1 Kelvin. It also interesting to know that the specific heat capacities in two other phases of water (i.e. ice and water vapour) are less than this. High specific heat of water is mainly due to the presence of a large number of hydrogen bonds between molecules of water. On a beach on a sunny day, it is noticed that the sand is often quite hot to walk on, but the water is always cool, even in the shallows. This is because sand has a lower specific heat capacity than that of water. Sand takes less energy to raise the temperature by one degree. Because water has a high specific heat capacity, it requires more energy to raise the temperature by one degree. Since the sun puts same rate of energy on water and sand, which heats up sand more quickly and water more slowly. Water covers around 70% of the Earth's surface and its high specific heat plays a very important role to sustain life in the earth. It is able to absorb a lot of heat without a significant rise in the temperature. When temperatures decrease, the heat which is stored is released, restraining a rapid drop in temperature. The combined effect is the buffering of heat. A relatively constant temperature without sudden rise and drop is essential to sustain life. Hence water is important for life.Q. Specific heat capacity of ice is ............... than and specific heat capacity of water vapour is ............... than that of water.

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If a gas has n degree of freedom, ratio of principal specific heats of the gas isa)1+ 2/nb)2nc)1 - 2/nd)-2nCorrect answer is option 'A'. Can you explain this answer?
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