Ideal gas equation is represented as `PV=nRT`. Gases present in universe were fond ideal in the Boyle's temperature range only and deviated more from ideal gas behavior at high pressure and low temperature.
The deviation are explained in term of compressibility factor `z`. For ideal behavior `Z=(PV)/(nRT)=1`. the main cause to show deviavtion were due to wrong assumptions made about forces oif attractions (which becomes significant at high pressure ) and volume `V` occupied by molecules in `PV=nRT` is supposed to be volume of gas or the volume of container in which gas is placed by assuming that gaseous molecules do not have appreciable volume. Actually volume of the gas is that volume in which each molecule of gas can move freely. If volume occupied by gaseous molecule is not negligible, then the term `V` would be replaced by the ideal volume which by available for free motion of each molecule of gas in 1 mole gas.
`V_("actual")=` volume of container -volume occupied by molecules
`=v-b`
Where `b` represent the excluded volume occupied by molecules present in one mole of gas.
Similarly for `n` mole gas
`V_("actual")=v-nb`
The ratio of coefficient of thermal expansion `alpha=(((delV)/(delT))_(P))/V` and the isothermal compressibility
`beta=-((delV)/(delP)_(T))` for an ideal gas is:

Options

A. `-P/T`
B. `P/T`
C. `T/P`
D. `-T/P`

Explanation

To solve the question regarding the ratio of the coefficient of thermal expansion (α) and isothermal compressibility (β) for an ideal gas, we will follow these steps:

### Step 1: Define the Coefficient of Thermal Expansion (α)
The coefficient of thermal expansion at constant pressure is defined as:
\[ \alpha = \left( \frac{\partial V}{\partial T} \right)_{P} \cdot \frac{1}{V} \]
where \(V\) is the volume of the gas.

### Step 2: Use the Ideal Gas Law
For an ideal gas, the ideal gas equation is given by:
\[ PV = nRT \]
where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature.

### Step 3: Differentiate Volume with Respect to Temperature
At constant pressure, we differentiate the ideal gas equation with respect to temperature \(T\):
\[ \frac{\partial V}{\partial T} = \frac{nR}{P} \]

### Step 4: Substitute into the Expression for α
Substituting this result into the expression for α:
\[ \alpha = \left( \frac{nR}{P} \right) \cdot \frac{1}{V} \]
Since from the ideal gas law \(PV = nRT\), we can express \(nR\) as \(PV\):
\[ \alpha = \frac{PV}{P} \cdot \frac{1}{V} = \frac{1}{V} \]

### Step 5: Define Isothermal Compressibility (β)
The isothermal compressibility is defined as:
\[ \beta = -\left( \frac{\partial V}{\partial P} \right)_{T} \cdot \frac{1}{V} \]

### Step 6: Differentiate Volume with Respect to Pressure
For an ideal gas, at constant temperature, we can rearrange the ideal gas law to find \(V\):
\[ V = \frac{nRT}{P} \]
Differentiating \(V\) with respect to \(P\):
\[ \frac{\partial V}{\partial P} = -\frac{nRT}{P^2} \]

### Step 7: Substitute into the Expression for β
Substituting this result into the expression for β:
\[ \beta = -\left( -\frac{nRT}{P^2} \right) \cdot \frac{1}{V} \]
Using \(V = \frac{nRT}{P}\):
\[ \beta = \frac{nRT}{P^2} \cdot \frac{P}{nRT} = \frac{1}{P} \]

### Step 8: Find the Ratio of α to β
Now, we can find the ratio of the coefficient of thermal expansion (α) to isothermal compressibility (β):
\[ \frac{\alpha}{\beta} = \frac{\frac{1}{V}}{\frac{1}{P}} = \frac{P}{V} \]

### Final Answer
Thus, the ratio of the coefficient of thermal expansion to the isothermal compressibility for an ideal gas is:
\[ \frac{\alpha}{\beta} = \frac{P}{V} \]