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Q1. What is meant by the term ' Molar specific heat ' of a gas?  The molar specific heat of hydrogen in the temperature range of about 250 K to 750 K is about (5/2) R.  At lower temperatures the value of molar specific heat of hydrogen decreases to the value typical of monoatomic gases (3/2) R while at higher temperatures, it tends to the value (7/2) R.  Explain.

Solution

Molar specific heat capacity of a gas refers to the amount of energy required for 1 mole of a substance to raise its temperature by 1 K.  In the temperature beyond 70 kelvin, rotational motion of H₂ gas starts.  So at 250 K < T < 750, the number of degrees of freedom becomes five -2 rotational and 3 translational Therefore,  C_v =f/2 R becomes, C_v = 5/2 R.  For lower temperatures only translational degrees of freedom will exist and no rotational freedom. Therefore,  C_v = 3/2 R.

Q2. What do you mean by an ideal gas?

Solution

An ideal gas or a perfect gas is that gas which strictly obeys the gas laws, such as Boyle's law, Charles's law etc.

Q3. In terms of the kinetic theory of gases, explain why the pressure of a gas in a closed container increases when the gas is heated?

Solution

As the temperature of a gas is increased, its pressure increases due to the following reasons : (i)  with an increase in temperature, the velocity of the gas molecules increases, and as such they strike the walls of the container more often than before. (ii)  Due to an increased velocity, each impact is more powerful than before.

Q4. The absolute temperature of a gas is made four times.  How many times will (a)  its total kinetic energy become? (b)  root mean square velocity of its molecules? (c)  How will pressure change?

Solution

(a)  kinetic energy will become four times. (b)  root mean-square velocity becomes twice. (c)  pressure is directly proportional to square of r.m.s velocity, hence pressure becomes four times.

Q5. What do you mean by the number of degrees of freedom of a dynamical system?  How many degrees of freedom are associated with (i)  a bob of an oscillating simple pendulum (ii)  an insect moving on a horizontal floor (iii)  a buzzing bee (iv)  a particle moving in a plane  

Solution

The number of degrees of freedom of a dynamical system is defined as the total number of independent quantities required to completely describe the position and configuration of the system. (i)  one  (ii) two  (iii)  three  (iv) two

Q6. The volume of a gas sample is increased.  Why does the pressure, which is exerted by the gas, decrease?

Solution

As the volume of a gas is increased, its pressure decreases because of the fact that : (i)  The molecules have to travel a longer distance between impact on the walls of the container. (ii)  These impacts are now distributed over a larger area.

Q7. (a)  How many degrees of freedom are there due to vibration motion of diatomic gas molecules? (b)  How many degrees of freedom are there for monoatomic, diatomic and triatomic gas due to translational and rotational motion only?

Solution

(a)  One.  Because diatomic gas has at the maximum six degrees of freedom (2 x 3 = 6) out of which, three are due to translational motion, two are due to rotational motion. (b)  Monoatomic gas has only three degrees of freedom and they are only translational. Diatomic gas has five degrees of freedom. Triatomic linear e.g. CO₂ gas has seven degrees of freedom and triatomic angular has six degrees of freedom.  

Q8. State two basic characteristics of the ideal gas.  Are real gases perfect gases?

Solution

The two basic characteristics of the ideal gas are: (i) The size of the molecule of a gas is zero. (ii) The molecules of a gas do not exert any force of attraction or repulsion amongst each other.  All real gases are not perfect gases, as they do not obey gas laws perfectly.

Q9. State law of equipartition of energy.

Solution

For any dynamical system in equilibrium, total energy is divided equally among all the degrees of freedom and energy associated with one degree of freedom is 1/2 kT.

Q10. When a gas is suddenly compressed, its temperature rises. Why?

Solution

This represents an adiabatic change. Work done in compressing the gas is converted into its internal energy and as such the temperature of the gas rises.