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प्रश्न
Derive Mayer’s relation.
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उत्तर
- Consider one mole of an ideal gas that is enclosed in a cylinder by a light, frictionless airtight piston.
- Let P, V, and T be the pressure, volume, and temperature respectively of the gas.
- If the gas is heated so that its temperature rises by dT, but the volume remains constant, then the amount of heat supplied to the gas (dQ1) is used to increase the internal energy of the gas (dE). Since the volume of the gas is constant, no work is done in moving the piston.
∴ dQ1 = dE = CV dT ..............(1)
where CV is the molar specific heat of the gas at constant volume. - On the other hand, if the gas is heated to the same temperature, at constant pressure, the volume of the gas increases by an amount say dV. The amount of heat supplied to the gas is used to increase the internal energy of the gas as well as to move the piston backward to allow expansion of gas. The work is done to move the piston dW = PdV.
∴ dQ2 = dE + dW = Cp dT ..............(2)
Where CP is the molar specific heat of the gas at constant pressure. - From equations (1) and (2),
∴ Cp dT = CV dT + dW
∴ (Cp - Cv)dT = PdV ..............(3) - For one mole of gas,
PV = RT
∴ P dV = R dT, since pressure is constant.
Substituting equation (3), we get
(Cp - Cv) dT = R dT
∴ Cp - Cv = R
This is known as Mayer’s relation between CP and CV. - Also, CP = M0SP and CV = M0SV, where M0 is the molar mass of the gas and SP and SV are respective principal specific heats. Thus, M0SP - M0SV = R/J
Where J is the mechanical equivalent of heat.
SP - Sv = `"R"/("M"_0"J")`
संबंधित प्रश्न
Given below are observations on molar specific heats at room temperature of some common gases.
| Gas |
Molar specific heat (Cv) (cal mol–1 K–1) |
| Hydrogen | 4.87 |
| Nitrogen | 4.97 |
| Oxygen | 5.02 |
| Nitric oxide | 4.99 |
| Carbon monoxide | 5.01 |
| Chlorine | 6.17 |
The measured molar specific heats of these gases are markedly different from those for monatomic gases. Typically, molar specific heat of a monatomic gas is 2.92 cal/mol K. Explain this difference. What can you infer from the somewhat larger (than the rest) value for chlorine?
Calculate the mass of ice required to lower the temperature of 300 g of water 40°C to water at 0°C.
(Specific latent heat of ice = 336 J/g, the Specific heat capacity of water = 4.2J/g°C)
A copper vessel of mass 100 g contains 150 g of water at 50°C. How much ice is needed to cool it to 5°C?
Given: Specific heat capacity of copper = 0.4 Jg-1 °C-1
The Specific heat capacity of water = 4.2 Jg-1 °C-1
The Specific latent heat of fusion ice = 336 Jg-1
Two metallic blocks P and Q of masses in ratio 2: 1 are given the same amount of heat. If their temperature rise by the same amount, compare their specific heat capacities.
Name the radiations for which the green house gases are transparent ?
Without green house effect, the average temperature of earth’s surface would have been:
(a) – 18℃
(b) 33℃
(c) 0℃
(d) 15℃
Read the following paragraph and answer the questions.
|
If heat is exchanged between a hot and cold object, the temperature of the cold object goes on increasing due to gain of energy and the temperature of the hot object goes on decreasing due to loss of energy. The change in temperature continues till the temperatures of both the objects attain the same value. In this process, the cold object gains heat energy and the hot object loses heat energy. If the system of both the objects is isolated from the environment by keeping it inside a heat resistant box (meaning that the energy exchange takes place between the two objects only), then no energy can flow from inside the box or come into the box. |
- Heat is transferred from where to where?
- Which principle do we learn about from this process?
- How will you state the principle briefly?
- Which property of the substance is measured using this principle?
What is the specific heat capacity of melting ice?
A solid of mass 80 g at 80°C is dropped in 400 g water at 10°C. If final temp. is 30°C, find the sp. heat cap. of the solid.
A piece of iron of mass 2.0 kg has a heat capacity of 966 J K-1. Find its specific heat capacity in S.I unit.
State the condition for the flow of heat energy from one body to another.
Write two advantages of high specific heat capacity of water.
Explain, why is water sprayed on roads in evening in hot summer?
Explain, why temperature in hot summer, falls sharply after a sharp shower?
1 kg of water freezes to form ice at 0°C. What amount of heat is withdrawn?
Calculate the amount of heat released when 5.0 g of water at 20°C is changed into ice at 0°C.
(Specific heat capacity of water = 4.2 J/g°C
Specific latent heat of fusion of ice = 336 J/g)
How much heat energy is necessary to raise the temperature of 5 kg of water from 20°C to 100°C?
Explain how heat capacity of a solid can be determined by the method of mixture.
Read the passage and answer the questions based on it.
If heat is exchanged between a hot and cold object, the temperature of the cold object goes on increasing due to gain of energy and the temperature of the hot object goes on decreasing due to loss of energy. The change in temperature continues till the temperatures of both objects attain the same value. In this process, the cold object gains heat energy and the hot object loses heat energy. If the system of both the objects is isolated from the environment by keeping it inside a heat-resistant box then no energy can flow from inside the box or come into the box. In this situation, we get the following principle.
Heat energy lost by the hot object = Heat energy gained by the cold object. This is called the ‘Principle of heat exchange’.
- Where does heat transfer take place?
- In such a situation which principle of heat do you perceive?
- How can this principle be explained in short?
- Which property of the substance is measured using this principle?
Two uniform brass rods A and B of length land 2l and radii 2r and r respectively are heated to the same temperature. The ratio of the increase in the volume ofB to that of A is ____________.
A monoatomic gas of pressure 'P' having volume 'V' expands isothermally to a volume '2V' and then adiabatically to a volume '16V'. The final pressure of the gas is ______.
(ratio of specific heats = `5/3`)
At same temperature and pressure of an ideal gas, ____________.
A piece of lead weighing 500 g gives out 1200 calories of heat when it is cooled from 100° C to 20° C. Find its specific heat.
To study energy exchange between hot and cold objects, the system of both objects is isolated from the environment by keeping them inside ______.
Two blocks P and Q of different metals having their mass in the ratio 2 : 1 are given same amount of heat. Their temperature rises by same amount. Compare their specific heat capacities.
Give one example where high specific heat capacity of water is used as heat reservoir.
In the method of mixtures, a hot metal is dropped into cold water and the mixture reaches a final temperature. Which principle is used to find the specific heat of the metal?
