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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?
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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?
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Define heat capacity.
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(g = 10 ms-2; Specific heat capacity of water = 4200 J / kg°C)
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(Specific latent heat of vaporization of steam = 2268 J/g, specific latent heat of fusion of ice = 336 J/f and specific heat capacity of water = 4.2 J/g°C)
Write the name.
The amount of heat absorbed at constant temperature by unit mass of a liquid to convert into gaseous phase.
Numerical Problem.
What is the heat in joules required to raise the temperature of 25 grams of water from 0°C to 100°C? What is the heat in Calories? (Specific heat of water = `(4.18"J")/("g"°"C")`
If 'f' is the number of degrees of freedom of a molecule of a gas and ratio of molar specific heats of a gas, ϒ = 1 + `2/"f"` where ϒ = Cp/Cv. The ratio of 'ϒ' for monoatomic gas to 'ϒ' for (rigid) f diatomic gas is ______.
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A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature (27°C). The ratio of specific heat of gases at constant volume respectively is:
