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Heat and Thermodynamics

Temperature

Degree of hotness or coldness of a body is called temperature. It is a quantity by which heat flows from one body to another when they are placed in contact to each other.

Heat

• Heat is the energy is transferred due to difference in temperature of the bodies.
• It is a form of energy.
• The temperature of a system increases if heat is supplied to or absorbed by it.
• Temperature of system decreases when heat is taken out of the system.
• Often, if the temperature changes by an exchange of heat, then $$ Q=mc (\Delta T) $$ where \(m = \text{mass}\), $c = \text{specific heat} \(T_2-T_1 = \Delta T = \text{change in temperature}\)
• The unit of heat in MKS is \(\text{Joule}\), in CGS, it is \(\text{erg}\) and in thermal units \(\text{calorie}\).
• \(4.2\text{Joule} = 1\text{calorie}\)

Some Terminologies

Specific Heat

The heat required to increase the temperature of unit mass of substance by \(1^oC\) is called specific heat of the substance. $$ Q = mc\Delta T $$ or $$ c = \frac{Q}{m\Delta T} $$

Thermal Equilibrium

Objects A and B are in contact. If heat flows from B, then A is at a higher temperature than B. When the heat flow from A to zero. the two objects are in thermal equilibrium.

Zeroth Law

When two systems A and B are separately in thermal equilibrium with a third system C, then the first two systems will also be in thermal equilibrium with each other It means if, $$ T_A=T_C\quad\text{and}\quad T_B = T_C\quad\text{then}\quad T_A = T_B $$ Thus according to this law, temperature is that intrinsic property of an object on the basis of which, we can say that whether the object is in thermal equilibrium with another or not

Conversion formulas of temperature scales:

1. $$ T_C=\frac{5}{9}(T_F - 32) $$
2. $$ T_F=\frac{9}{5}T_C+32 $$
3. $$ T_K=273 + T_C $$
4. $$ T_K= 273 + \frac{5}{9}(T_F-32) $$
5. $$ T_C= T_K - 273 $$
6. $$ T_F= \frac{9}{5}(T_K - 273) + 32 $$

Relation between \(^oC\). \(^oF\) and \(K\) Temperature:

\[ \frac{T_C-0}{100} = \frac{T_F-32}{180} = \frac{T_K-273}{100} \]

Scale	Lower Point	Upper Point	Number of divisions
Celsius	\(0^oC\)	\(100^oC\)	\(100\)
Fahrenheit	\(32^oF\)	\(212^oF\)	\(180\)
Kelvin	\(273K\)	\(373K\)	\(100\)

Do you know?

The centigrade and Fahrenheit scale shows the same reading at temperature of \(-40^o\).

Do you know?

Celsius degree is larger than a Fahrenheit degree by a factor of \(9/5\).

Kinetic Molecular Theory of Gasses

Main points of Kinetic Molecular Theory of Gasses are given as:

1. Small volume contains large number of molecules.
2. Molecules move randomly and do not exert force on one another except when they collide.
3. Molecules collide each other elastically.
4. The collisions with 'walls' give rise to gas pressure.
5. Gravity does not affect the molecular motion.
6. Volume of gas molecules is negligible as compared to the actual volume of the gas.

Expressions for Pressure of Gas

It is derived on the basis of the kinetic theory that the pressure of a gas is given by $$P=\frac{1}{3}\frac{mn}{V}\lt v^2\gt\quad\text{or}\quad P=\frac{2}{3}N_o\lt\frac{1}{2}mv^2\gt $$

where \(m\quad\;\;\;=\) Mass of each molecule
\(\quad\quad n \quad\quad\;\;=\) Total number of molecules in the gas
\(\quad\quad V \quad\quad\;=\) Volume of the gas
\(\quad\quad\lt v^2\gt \;=\) Mean-square velocity of the molecules
\[ \quad\quad N_o \quad\quad= \frac{n}{V} \]

• Since \(mn\) is the mass of whole gas and \(V\) is the its volume, so $$ \frac{mn}{V} $$ is the density \(\rho\) of the gas. Thus $$ P = \frac{1}{3} \rho \lt v^2\gt $$
• We can write this expression as $$ P=\frac{1}{3}\rho\lt v^2\gt=\frac{2}{3}\left(\frac{1}{2}\rho\lt v^2\gt\right) = \frac{2}{3}KE $$ where \(KE\) represents the kinetic energy of the gas per unit volume. Hence $$ P=\frac{2}{3}KE $$. Thus, the pressure of a gas is equal to two-third of its kinetic energy per unit volume.

Expressions for Temperature of gas

Consider \(1\;mole\) of a gas at absolute temperature \(T\) and occupying volume \(V\). Then pressure of \(1\;mole\) of gas is given by:

\[ P =\frac{1}{2}\frac{mN}{V} \lt v^2\gt \]

• Where \(N\) is Avogadro's number i.e., \(N(=6.02\times10^{23})\) is the number of molecules in \(1\;mole\) of gas. \(mN\) = Mass of 1gm-molecule of the gas Molecular weight M

\[ PV=\frac{1}{2}M \lt v^2\gt \]

• Using ideal gas equation \(PV = RT\) where \(R\) is gas constant, \(n = 1\) we have \[RT=\frac{1}{3}M\lt v^2\gt\] or \[\sqrt{\lt v^\gt}=\sqrt{\frac{3RT}{M}}\]

• \(\sqrt{\lt v^\gt}\) is the square-root of the mean-square velocity of the gas molecules. It is called root-mean-square velocity (rms velocity) Thus $$ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} $$ Also $$ v_{\text{rms}}\propto \sqrt{T} $$

• Thus, the root-mean-square velocity of the molecules of a gas is directly proportional to the square-root of the absolute temperature of the gas.

Relation between RMS Velocity and Molecular Weight:

\[ (v_1)_\text{rms}=\sqrt{\frac{3RT}{M_1}} \]

and

\[ (v_2)_\text{rms}=\sqrt{\frac{3RT}{M_2}} \]

so

\[ \frac{(v_1)_\text{rms} }{(v_2)_\text{rms}} = \sqrt{\frac{M_2}{M_1}} \]

Relation between Temperature and Kinetic Energy of Molecules:

Kinetic energy of 1 gm-molecule of the gas

\[ KE = \frac{1}{2} M \lt v^2 \gt = \frac{1}{2}M\left(\frac{3RT}{M}\right) = \frac{3}{2} RT \]

\(M=\)Molecular weight of gas
\(<v^2> =\) Mean-square velocity of molecules
\(R =\) Universal Gas constant

• Since there are \(N\) molecules, so the average kinetic energy of one molecule $$=\frac{3}{2}\frac{RT}{N}=\frac{3}{2}\left(\frac{R}{N}\right) T $$
• The ratio \(R/N\) is Boltzmann's constant \(k\)
• Average kinetic energy of one molecule $$ = \frac{3}{2kT} $$

Gas Laws

Boyle's Law:

As we know that $$ PV =\frac{2}{3}N \lt \frac{1}{2} mv^2\gt $$ At constant temperature, \(<K.E>\) remains constant, so the right hand side of th3 equation is constant. Hence \(PV =\)Constant

or $$ P\propto\frac{1}{V} $$ Thus pressure \(P\) is inversely proportional to volume \(V\) at constant temperature of the gas which is Boyle's law.

Charles' Law:

As $$ V=\frac{2}{3}\frac{N}{P}\lt \frac{1}{2}mv^2\gt p $$

If pressure is kept constant

\[ V\propto \lt\frac{1}{2} mv^2\gt \]

\[ \lt\frac{1}{2} mv^2\gt\propto T \]

\[ V\propto T \]

Thus volume is directly proportional to absolute pressure is constant. This is known as Charles' law.

Specific Heat

Amount of heat required to raise the temperature of a substance through \(1K\) called heat capacity, denoted by \(C\). $$ Q=Cm\Delta T $$

• Specific heat is the amount of heat required to raise the temperature of unit mass through unit temperature, $$ Q = C_{Sp}\Delta T $$

Do you know?

Molar specific heat of diatomic gas is greater than that of monoatomic gas.

Avagadro's law states:

Equal volumes of gases at same temperature and pressure contain equal number of molecules.

Law of beat exchange states:

If no heat is lost to surroundings or gained from it, then \(\text{Heat lost} = \text{Heat gained}\)

Thermodynamic

• Thermodynamics is the study of the relationship between heat and other forms of energy.
• Thermodynamic states describe the state of a system.
• Any collection of matter having distinct boundaries is called a system.
• Boyle's law states that at constant temperature.
  \(PV=\)constant or \(P_1V_1=P_2V_2\)
• Charle's law states that at constant pressure
  \[\frac{V}{T} =\]constant or \[\frac{V_1}{T_1}=\frac{V_2}{T_2}\]
• Sum of translational K.E, rotational K.E vibrational K.E and RE due to intermolecular forces is called internal energy.
• Internal energy of an ideal gas system is the translational K.E of its molecules.
• Increase in temperature of the object is an indication of increase in the internal energy.
• Thermodynamic work at constant pressure is \(W= P\Delta V\), and if pressure is not constant then $$ W = \sum^{n}_{\Delta V_i\to 0} P_i\Delta V_i \quad \text{where} \quad n\to\infty $$

Laws of Thermodynamic

To be completed...

First Law When heat is transformed into other terms or energ■ total heat remains ::constant_ AQ = AU + AW Where /IQ is +ve when heat is added and vice versa AW is -ve when work is done by system and vice versa. tnfe--_acersfrom 1St Law of Thermodynamics a

AU --= AQ - AW range 117 internal Ilea! energy flowing au/ energv ] = (Heal energv flawing in )- aN mechanical work Internal energy is a state function. i.e. depends on initial and final states For a cyclic process, we have-\ti--,-0 , U, = Ut , ,\Q' = AW eat , ilions of 1^st law of Thermodynamics Isothermall'fCSAI is that in which temperature remains constant. \Q ' \AI as AU - 0 i. -r11.9ric process is that in which volume remains constant. f': AU as AW = 0 -§';)P__.krjsprs)cess is that in which pressure remains constant, 'TRY TEST SERIES 141

To be completed...

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