lec3

Clausius-Clapeyron Equation
The equilibrium between water and water vapor depends upon the temperature of the system. If the temperature increases the saturation pressure of the water vapor increases. The rate of increase in vapor pressure per unit increase in temperature is given by the Clausius-Clapeyron equation. Let p be the saturation vapor pressure and T the temperature. The Clausius-Clapeyron equation for the equilibrium between liquid and vapor is then the relationship between the vapor pressure of a liquid and the temperature is :
The linear relationship can be expressed mathematically as
lnP=(-?H Vap)/R [1/T]+C (1)
where ln P is the natural logarithm of the vapor pressure , ?Hvap is the heat of vaporization, R is the universal gas constant (8.31 J•K-1mol-1), T the absolute temperature, and C a constant.
This is the Clausius-Clapeyron equation, which gives us a way of finding the heat of vaporization, the energy that must be supplied to vaporize a mole of molecules in the liquid state.

A plot of ln P vs. 1/T has the form of a straight line. Com¬pare equation (1) to
y = mx + b (2)
which is the equation for a straight line. In equation (1), ln P is y, 1/T is x, and -?Hvap/R is m.


Question :
The two point version of the Clausius-Clapeyron equation is
ln P2/P1=??Hvap/R[1/T2?1/T1]
Use this equation and the following data to calculate ?Hvap of water: At 20 °C, the VP of water is 17.5 Torr. At 60°C, the VP of water is 149.4 Torr (Use R = 8.314 J•K-1•mol-1).

Real gas – Van der Waals equation:
There are attractive forces between real molecules, which reduce the pressure: p ? wall collision frequency and p ? change in momentum at each collision. Both factors are proportional to concentration, n/V, and p is reduced by an amount a(n/V)2, where a depends on the type of gas. [Note: a/V2 is called the internal pressure of the gas]. Real gas molecules do attract one another
(P+n²a/V²)(V-nb)=nRT
OR
P=RT/(Vm-b)-(a/Vm)²
a is also different for different gases
a: describes attractive force between pairs of molecules. Goes as square of the concentration (n/V)2
where n is the mole number, a and b are constants characteristic of a particular gas, and R the gas constant. P, V, and T are as usual the pressure, volume, and temperature.
Enthalpy
For a system that changes volume, the internal energy is not equal to the heat supplied, as for a fixed volume system some energy supplied as heat to the system returns to surroundings as expansion work:
dU < dq, because dU = dq + dw
When heat is supplied to the system at a constant pressure (e.g., reaction containers open to atmosphere), another thermodynamic state function known as enthalpy, H, can be measured accurately:
H = U + pV
The change in enthalpy is equal to heat supplied to the system at constant pressure
dH = dq
1. Infinitesimal change in state of the system: U changes to U + dU, p changes to p + dp, V changes to V + dV, so H = U + PV becomes
H + dH = (U + dU) + (p + dp) (V + dV)
= U + dU + pV + pdV + Vdp + dp dV
2. The product of two infinitesimal quantities, dp dV, disappears.
Since H = U + PV, we have written
H +dH = H + dU + pdV + Vdp
dH= dU + pdV + Vdp
3. Substitute in dU = dq + dw
dH = dq + dw + pdV + Vdp
4. System is in mechanical equilibrium with surroundings at pressure p, so there is only expansion work (dw = -pdV)
dH = dq + Vdp
5. Impose condition that heating is done at constant pressure, so dp = 0
dH =dq (constant p)

Compressibility factor
Compression: The way in which the volume of a material decreases with pressure at constant temperature is described by the isothermal compressibility, ?:

Later, we shall need to distinguish between “isothermal compressibility” and “adiabatic compressibility”, and we shall need a subscript to the symbol ?a in order to distinguish between the two. For the time being, however, ? with no subscript will be taken to mean the isothermal compressibility.
The reciprocal of ? is called the isothermal bulk modulus, sometimes called the isothermal incompressibility.
Coefficient of expansion
In an ideal world, we’d use ?, b, ? respectively, for the coefficients of linear, area and volume expansion.
Coefficient of linear expansion: ?
Coefficient of area expansion: b
Coefficient of volume expansion: ?
For small ranges of temperature, the increases in length, area and volume with temperature can be represented by
l2 = l1[1 + ? (T2 ? T1)]
A2 = A1 [1 + b (T2 ? T1)]
V2 = V1[1 + ?(T2 ? T1)].

Here ?, b, ? are the approximate coefficients of linear, area and volume expansion respectively over the temperature range T1 to T2. For all three, the units are degree?1 that is C? ?1 or K?1.

?=1/L(?L /?T)P
b=1/A(?A/?T)P
?=1/V(?V/?T)P
The relations b = 2? and ? = 3? are exact.
For solids, the coefficient of linear expansion is usually the appropriate parameter; for liquids and gases the volume coefficient is usually appropriate. For most familiar common metals the coefficient of linear expansion is of order 10?5 K?1. .
At room temperatures and above, the coefficient of linear expansion of metals doesn’t vary a huge amount with temperature, but at low temperatures the coefficient of expansion varies much more rapidly with temperature – and so does the specific heat capacity.
Boyle’s Law
This law was formulated by Robert Boyle in the year 1662. It states that “The absolute pressure of a given mass of a perfect gas varies inversely as its volume, when the temperature remains constant”. P ?1 / V or PV = constant The more useful form of the above equation is P1V1 = P2V2 = P3V3 = ……… = Constant Mathematically, P? 1/ V or PV = Constant, if temperature remains constant. If the gas changes from sate (1) to sate (2) at constant temperature, the two end states are related by the equation : P1V1 = P2V2
Charle’s Law
This Law was formulated by a French scientist Jacques A.C. Charles in about 1787. it may be stated as “The volume of a given mass of a perfect gas varies directly as its absolute temperature, when the absolute pressure remains constant.” Mathematically, V ?T or V/T = Constant. (or) V1 = V2 = V3 = Constant T1 T2 T3. The Charles’s law can also be stated as follows : “ If the volume of a gas is kept constant, then the absolute pressure is directly proportional to the absolute temperature.” At constant volume, P?T or P/T = Constant At constant volume P1/T1 = P2/T2 The change of states at constant pressure and constant volume are represented on P-V diagrams.
Avogadro’s Law It states,
“Equal volumes of all gases at the same temperature and pressure, contain equal number of molecules. Thus, according to Avogadro’s law. 1m3 of Oxygen (O2) will contain the same number of molecules as 1m3 of Hydrogen (H2 ) when the Temperature and pressure remain same.?
Joule’s Law
It states, “The change of Internal energy of a perfect gas is directly proportional to the change of temperature.”
Energy and Work
There are many different types of energy, and different formulas for calculating it, but they should all be measured with the same unit, the Joule.
The following are a few different mathematical definitions of energy. Kinetic Energy = ½ mv2 where m is mass and v is speed (or velocity without direction). Kinetic Energy is the energy of motion – anything that is moving has kinetic energy. The higher the speed, the more kinetic energy an object has. Gravitational Potential Energy = m g h where m is mass, g is gravitational acceleration, and h is the height of the object relative to a reference point (e.g. the ground). The higher something is above the ground, the more potential it has to fall, increase speed, and thus gain kinetic energy.
Work is defined as the change in the level of energy of a system, so it is also measured in Joules. Mechanical work can be measured by this formula: W = F.D ., where F is the force exerted and, and D is the distance over which the force was exerted. So, if I apply a steady force of 100 N on an object, and the object moves 5 m, then the amount of work I did on the object is: W= 100 × 5 = 500 J
Power
The concept of power follows directly from work. Power is simply the rate at which work is done and is measured in Watts
REVERSIBLE AND IRREVERSIBLE PROCESS
A reversible process is defined as a process that can be reversed without leaving any trace on the surroundings. It means both system and surroundings are returned to their initial states at the end of the reverse process. Reversible processes do not occur and they are only idealizations of actual processes. We use reversible process concept because, a) they are easy to analyze (since system passes through a series of equilibrium states); b) they serve as limits (idealized models) to which the actual processes can be compared.
Some factors that cause a process to become irreversible:
• Friction
• Unrestrained expansion and compression
• Mixing
• Heat transfer (finite ?T)
• Inelastic deformation
• Chemical reactions
In a reversible process things happen very slowly, without any resisting force, without any space limitation ? everything happens in a highly organized way (it is not physically possible ? it is an idealization). Internally reversible process: if no irreversibilities occur within the boundaries of the system. In these processes a system undergoes through a series of equilibrium states, and when the process is reversed, the system passes through exactly the same equilibrium states while returning to its initial state.
Externally reversible process: if no irreversibilities occur outside the system boundaries during the process. Heat transfer between a reservoir and a system is an externally reversible process if the surface of contact between the system and reservoir is at the same temperature.
Totally reversible (reversible): both externally and internally reversible processes. Examples: Some examples of nearly reversible processes are:
Frictionless relative motion.
Expansion and compression of spring.
Frictionless adiabatic expansion or compression of fluid.
Polytropic expansion or compression of fluid.
Isothermal expansion or compression.
Electrolysis.
An irreversible process is one in which heat is transferred through a finite temperature. In summary, processes that are not reversible are called irreversible.
Examples of irreversible process
Relative motion with friction
Combustion
Diffusion
Free expansion
Throttling
Electricity flow through a resistance
Heat transfer (viii) Plastic deformation.
Adiabatic process
An adiabatic process takes place when no thermal energy enters or leaves the system. This occurs if the system is perfectly insulated or if the process occurs so rapidly that there is no heat transfer.