In this section, we will focus only on mechanical work, i.e. pressure-volume work.

Let us assume a cylinder that contains one mole of an ideal gas. The cylinder is fitted with a piston. Total volume of gas is V_{i} and pressure of the gas inside is p. If external pressure (p_{ex}) is greater than p, the piston will move inward till the pressure inside becomes equal to p_{ex}. Let us assume that this change is achieved in a single step, and final volume becomes V_{f}. During the compression, piston moves a distance (l) and area of cross section of piston is A.

Volume change `=l×A=ΔV=V_f-V_i`

We know that pressure = force ÷ area

Or, Force = Pressure × Area

So, force on piston = p_{ex} . A

We also know that work = force × displacement

So, `w=p_ex×A×l`

Fig Ref: NCERT Book

If w is the work done on the system by movement of piston then

w = force × distance

`=p_(ex)×A×l`

`=p_(ex)(-ΔV)`

`=-p_(ex)ΔV`

`=-p_(ex)(V_f-V_i)` …………………(2)

The negative sign shows that in case of compression, work being done on the system, (V_{f} - V_{i} will be negative and negative multiplies by negative will be positive. So, the sign obtained for work will be positive.

Expansion of a gas in vacuum (p_{ex} = 0) is called free expansion. No work is done during free expansion of an ideal gas.

For isothermal (T = constant) expansion of an ideal gas into vacuum: w = 0 because p_{ex} = 0.

The equation, ΔU = q + w can be expressed for isothermal irreversible and reversible changes as follows:

**For isothermal irreversible change:**

`q=-w=p_(ex)(V_f-V_i)`

**For isothermal reversible change:**

`q=-w=nR\T In (V_f)/(V_i)`

= 2.303 nRT log `(V_f)/(V_i)`

For adiabatic change, q = 0

ΔU = w_{ad}

Heat absorbed at constant volume is equal to change in internal energy. But most of chemical reactions are carried out not at constant volume, but in flasks and test tubes at constant atmospheric pressure. So, we need to define another state function for these conditions.

ΔU = q_{p} - pΔV

For initial and final states, the equation can be written as follows:

`U_2-U_1=q_p-p(V_2-V_1)`

Or, `q_p=(U_2+pV_2)-(U_1+pV_1)` ……………(3)

Now , another thermodynamic function, the enthalpy H can be defined as follows:

`H=U+pV`

So, equation (3) can be written as follows:

`q_p=H_2-H_1=ΔH`

H is a state function because it depends on U, p and V but is independent of path. So, q_{p} is also path independent.

For constant p:

ΔH = ΔU = pΔV

ΔH is negative for exothermic reactions, and it is positive for endothermic reactions.

ΔH = ΔU – q_{V} ………(4)

Usually, the difference between ΔH and ΔU is not significant for systems consisting of only solids and/or liquids because solids and liquids do not suffer significant volume changes upon heating. But difference becomes significant when gases are involved.

Let us assume a reaction involving gases. At constant pressure and temperature, the ideal gas equation can be written as follows:

`pV_A=n_A\RT`

`p_V_B=n_B\RT`

Or, `p(V_B-V_A)=(n_B-n_A)RT`

Or, `pΔV=Δn_g\RT` ……….(5)

Here, V_{A} is total volume of gaseous reactants, V_{B} is total volume of gaseous products, n_{A} is number of moles of gaseous reactants and n_{B} is number of moles of gaseous products.

Substituting the value of pΔV in equation (4) we get:

ΔH = ΔU + Δn_{g} RT ………(6)

**Extensive Property:** A property whose value depends on the quantity or size of matter present in the system is called extensive property. Examples: mass, volume, internal energy, enthalpy, heat capacity, etc.

**Intensive Property:** A property which does not depend on the quantity or size of the matter is called intensive property. Examples: temperature, pressure, density, etc.

The increase in temperature is proportional to the heat transferred.

q = coeff × ΔT

this equation can also be written as follows:

`q=CΔT`

The coefficient, C is called the heat capacity. The magnitude of heat capacity depends on size, composition and nature of the system.

**Molar Heat Capacity:** The heat capacity of one mole of the substance is called molar heat capacity. In other words, the quantity of heat required to raise the temperature of one mole by one degree Celsius (or one Kelvin) is called molar heat capacity. Molar heat capacity is expressed by following equation:

`C_m=C/n`

**Specific Heat:** The quantity of heat required to raise the temperature of one unit mass of a substance by one degree Celsius (or one Kelvin) is called specific heat or specific heat capacity of that substance. The required heat q, to raise the temperature of a sample can be calculated as follows:

`q=c×m×ΔT=CΔT`

Where, c is specific heat, m is mass and ΔT is change in temperature.

Relationship between C_{p} and C_{V} for an ideal gas:

C_{V} is heat capacity at constant volume, and C_{p} is heat capacity at constant pressure.

**At constant volume:** `q_V=C_VΔT=ΔU`

**At constant pressure:** `q_p=C_pΔT=ΔH`

For one mole of an ideal gas:

`ΔH=ΔU+Δ(pV)`

`=ΔU+Δ(RT)`

`=ΔU+RΔT`

Or, `ΔH=ΔU+RΔT` ………………(7)

On putting the value of ΔH and ΔU, we have

`C_pΔT=C_VΔT+RΔT`

Or, `C_p=C_V+R`

Or. `C_p-C_V=R` ………(8)

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