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Degrees of Freedom The number of degrees of freedom of a dynamical system is defined as the total number of co-ordinates or independent variables required to describe the position and configuration of the system. For translatory motion (a) A particle moving in a straight line along any one of the axes has one degree of freedom (e.g). Bob of an oscillating simple pendulum. (b) A particle moving in a plane (X and Y axes) has two degrees of freedom. (e.g) An ant that moves on a floor. (c) A particle moving in space (X, Y and Z axes) has three degrees of freedom. (e.g) a bird that flies. A point mass cannot undergo rotation, but only translatory motion. A rigid body with finite mass has both rotatory and translatory motion. The rotatory motion also can have three co-ordinates in space, like translatory motion ; Therefore a rigid body will have six degrees of freedom ; three due to translatory motion and three due to rotatory motion. Monoatomic molecule Since a monoatomic molecule consists of only a single atom of point mass it has three degrees of freedom of translatory motion along the three co-ordinate axes as shown in figure. Examples : molecules of rare gases like helium, argon, etc. Diatomic molecule The diatomic molecule can rotate about any axis at right angles to its own axis. Hence it has two degrees of freedom of rotational motion in addition to three degrees of freedom of translational motion along the three axes. So, a diatomic molecule has five degrees of freedom as shown in figure. Examples: molecules of O_{2}, N_{2}, CO, Cl_{2}, etc. Triatomic molecule (Linear type) In the case of triatomic molecule of linear type, the centre of mass lies at the central atom. It, therefore, behaves like a diamotic moelcule with three degrees of freedom of translation and two degrees of freedom of rotation, totally it has five degrees of freedom as shown in figure. Examples: molecules of CO_{2}, CS_{2}, etc. Triatomic molecule (Non-linear type) A triatomic non-linear molecule may rotate, about the three mutually perpendicular axes, as shown in figure. Therefore, it possesses three degrees of freedom of rotation in addition to three degrees of freedom of translation along the three co-ordinate axes Hence it has six degrees of freedom Examples : molecules of H_{2}O, SO_{2}, etc. In all the above cases, only the translatory and rotatory motion of the molecules have been considered. The vibratory motion of the molecules has not been taken into consideration. Law of equipartition of energy Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom. The energy associated with each degree of freedom per moelcule is ½ kT, where k is the Boltzmann’s constant. Let us consider one mole of a monoatomic gas in thermal equilibrium at temperature T. Each molecule has 3 degrees of freedom due to translatory motion. According to kinetic theory of gases, the mean kinetic energy of a molecule is 3/2 kT. ½ mC^{2} = ½ mC_{x}^{2} + ½ mC_{y}^{2 }+ ½ mC_{z}^{2} So, ½ mC_{x}^{2} + ½ mC_{y}^{2 }+ ½ mC_{z}^{2} = 3/2 kT Since molecules move at random, the average kinetic energy correspoonding to each degree of freedom is the same. ½ mC_{x}^{2} = ½ mC_{y}^{2} = ½ mC_{z}^{2} That is, ½ mC_{x}^{2} = ½ mC_{y}^{2} = ½ mC_{z}^{2} = ½ kT Thus, mean kinetic energy per molecule per degree of freedom is ½ kT. Thermal equilibrium Let us consider a system requiring a pair of independent co-ordinates X and Y for their complete description. If the values of X and Y remain unchanged so long as the external factors like temperature also remains the same, then the system is said to be in a state of thermal equilibrium. Two systems A and B having their thermodynamic co-ordinates X and Y and X_{1} and Y_{1} respectively separated from each other, for example, by a wall, will have new and common co-ordinates X^{′} and Y^{′ }spontaneously, if the wall is removed. Now the two systems are said to be in thermal equilibrium with each other. Zeroth law of thermodynamics If two systems A and B are separately in thermal equilibrium with a third system C, then the three systems are in thermal equilibrium with each other. Zeroth law of thermodynamics states that two systems which are individually in thermal equilibrium with a third one, are also in thermal equilibrium with each other. This Zeroth law was stated by Flower much later than both first and second laws of thermodynamics. This law helps us to define temperature in a more rigorous manner. Temperature If we have a number of gaseous systems, whose different states are represented by their volumes and pressures V_{1}, V_{2}, V_{3} ... and P_{1}, P_{2}, P_{3}... etc., in thermal equilibrium with one another, we will have φ_{1} (P_{1},V_{1}) = φ_{2} (P_{2}, V_{2}) = φ_{3} (P_{3}, V_{3}) and so on, where φ is a function of P and V. Hence, despite their different parameters of P and V, the numerical value of the these functions or the temperature of these systems is same. Temperature may be defined as the particular property which determines whether a system is in thermal equilibrium or not with its neighbouring system when they are brought into contact. Refer this video to know more about on, “Degrees of Freedom”. Problem (JEE Advanced): Hydrogen gas is heated in a vessel to a temperature of 1000 K. Let each molecule possess an energy E_{1}. A few molecules escape into atmosphere at 400 K. Due to collisions their energy changes to E_{2}. Calculate E_{1}/E_{2}. Solution: At 1000 K, degree of freedom of hydrogen (diatomic gas), n = 7. Thus, K.E. per molecule, E_{1} = 7/2 kT = 7/2 k (1000) = 3500 k At 400 K, degree of freedom of H_{2} = 5 K.E. per molecule, E_{2} = 5/2 kT = 5/2 k (400) = 1000 k Therefore, E_{1}/E_{2} = 3500 k/1000 k = 7/2
The number of degrees of freedom of a dynamical system is defined as the total number of co-ordinates or independent variables required to describe the position and configuration of the system.
For translatory motion
(a) A particle moving in a straight line along any one of the axes has one degree of freedom (e.g). Bob of an oscillating simple pendulum.
(b) A particle moving in a plane (X and Y axes) has two degrees of freedom. (e.g) An ant that moves on a floor.
(c) A particle moving in space (X, Y and Z axes) has three degrees of freedom. (e.g) a bird that flies.
A point mass cannot undergo rotation, but only translatory motion. A rigid body with finite mass has both rotatory and translatory motion. The rotatory motion also can have three co-ordinates in space, like translatory motion ; Therefore a rigid body will have six degrees of freedom ; three due to translatory motion and three due to rotatory motion.
Since a monoatomic molecule consists of only a single atom of point mass it has three degrees of freedom of translatory motion along the three co-ordinate axes as shown in figure.
Examples : molecules of rare gases like helium, argon, etc.
The diatomic molecule can rotate about any axis at right angles to its own axis. Hence it has two degrees of freedom of rotational motion in addition to three degrees of freedom of translational motion along the three axes. So, a diatomic molecule has five degrees of freedom as shown in figure. Examples: molecules of O_{2}, N_{2}, CO, Cl_{2}, etc.
In the case of triatomic molecule of linear type, the centre of mass lies at the central atom. It, therefore, behaves like a diamotic moelcule with three degrees of freedom of translation and two degrees of freedom of rotation, totally it has five degrees of freedom as shown in figure. Examples: molecules of CO_{2}, CS_{2}, etc.
A triatomic non-linear molecule may rotate, about the three mutually perpendicular axes, as shown in figure. Therefore, it possesses three degrees of freedom of rotation in addition to three degrees of freedom of translation along the three co-ordinate axes Hence it has six degrees of freedom Examples : molecules of H_{2}O, SO_{2}, etc.
In all the above cases, only the translatory and rotatory motion of the molecules have been considered. The vibratory motion of the molecules has not been taken into consideration.
Law of equipartition of energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom. The energy associated with each degree of freedom per moelcule is ½ kT, where k is the Boltzmann’s constant.
Let us consider one mole of a monoatomic gas in thermal equilibrium at temperature T. Each molecule has 3 degrees of freedom due to translatory motion. According to kinetic theory of gases, the mean kinetic energy of a molecule is 3/2 kT.
½ mC^{2} = ½ mC_{x}^{2} + ½ mC_{y}^{2 }+ ½ mC_{z}^{2}
So, ½ mC_{x}^{2} + ½ mC_{y}^{2 }+ ½ mC_{z}^{2} = 3/2 kT
Since molecules move at random, the average kinetic energy correspoonding to each degree of freedom is the same.
½ mC_{x}^{2} = ½ mC_{y}^{2} = ½ mC_{z}^{2}
That is, ½ mC_{x}^{2} = ½ mC_{y}^{2} = ½ mC_{z}^{2} = ½ kT
Thus, mean kinetic energy per molecule per degree of freedom is ½ kT.
Let us consider a system requiring a pair of independent co-ordinates X and Y for their complete description. If the values of X and Y remain unchanged so long as the external factors like temperature also remains the same, then the system is said to be in a state of thermal equilibrium.
Two systems A and B having their thermodynamic co-ordinates X and Y and X_{1} and Y_{1} respectively separated from each other, for example, by a wall, will have new and common co-ordinates X^{′} and Y^{′ }spontaneously, if the wall is removed. Now the two systems are said to be in thermal equilibrium with each other.
If two systems A and B are separately in thermal equilibrium with a third system C, then the three systems are in thermal equilibrium with each other. Zeroth law of thermodynamics states that two systems which are individually in thermal equilibrium with a third one, are also in thermal equilibrium with each other.
This Zeroth law was stated by Flower much later than both first and second laws of thermodynamics. This law helps us to define temperature in a more rigorous manner.
If we have a number of gaseous systems, whose different states are represented by their volumes and pressures V_{1}, V_{2}, V_{3} ... and P_{1}, P_{2}, P_{3}... etc., in thermal equilibrium with one another, we will have φ_{1} (P_{1},V_{1}) = φ_{2} (P_{2}, V_{2}) = φ_{3} (P_{3}, V_{3}) and so on, where φ is a function of P and V. Hence, despite their different parameters of P and V, the numerical value of the these functions or the temperature of these systems is same.
Temperature may be defined as the particular property which determines whether a system is in thermal equilibrium or not with its neighbouring system when they are brought into contact.
Hydrogen gas is heated in a vessel to a temperature of 1000 K. Let each molecule possess an energy E_{1}. A few molecules escape into atmosphere at 400 K. Due to collisions their energy changes to E_{2}. Calculate E_{1}/E_{2}.
At 1000 K, degree of freedom of hydrogen (diatomic gas), n = 7.
Thus, K.E. per molecule, E_{1} = 7/2 kT = 7/2 k (1000) = 3500 k
At 400 K, degree of freedom of H_{2} = 5
K.E. per molecule, E_{2} = 5/2 kT = 5/2 k (400) = 1000 k
Therefore,
E_{1}/E_{2} = 3500 k/1000 k = 7/2
A degree of freedom of a physical system refers to a (typically real) parameter that is necessary to characterize the state of a physical system.
In mechanics, a point particles state at any given time can be described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.
In statistical mechanic, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space.
A degree of freedom may be any useful property that is not dependent on other variables. For example, in the 3D ideal chain model, two angles are necessary to describe each monomer's orientation.
In statistical mechanics and thermodynamics, it is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic way to the energy of the system. They are also variables that contribute quadratically to the Hamiltonian.
Degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space and degrees of freedom are sometimes referred to as its dimensions.
According to theorem of equipartition of energy, the energy of a system in thermal equilibrium is equally divided among all degrees of freedom.
Each degree of freedom contributes the same amount of average energy to the total, ½ k_{B}T per molecule.
The internal energy of a perfect gas is
(a) partly kinetic and partly potential
(b) wholly potential
(c) wholly kinetic
(d) depends on the ratio of two specific heats
The translational kinetic energy of gas molecules for one mole of the gas is equal to :
(a) 3/2 RT (b) 2/3 kT
(c) ½ RT (d) 3/2 kT
For an ant moving on the horizontal surface, the number of degrees of freedom of the ant will be:
(a) 1 (b) 2
(c) 3 (d) 6
At a given temperature, the ratio of the RMS velocity of hydrogen to the RMS velocity of oxygen is
(a) 4 (b) ¼
(c) 16 (d) 8
First law of thermodynamics is a consequence of the conservation of
(a) momentum (b) charge
(c) mass (d) energy
c
a
b
d
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