If P, V, M, T and R are symbols of pressure, volume, molecular weight, temperature and Gas contstant, what is the equation of density of ideal gas

Options

A. `(RT)/(pM)`
B. `(p)/(RT)`
C. `(M)/(V)`
D. `(pM)/(RT)`

Explanation

To derive the equation for the density of an ideal gas using the given symbols, we can follow these steps:

### Step-by-Step Solution:

1. **Start with the Ideal Gas Law:**
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

**Hint:** Remember that the ideal gas law relates pressure, volume, temperature, and the number of moles of gas.

2. **Rearrange the Ideal Gas Law to Solve for Volume:**
We can rearrange the equation to find the volume \( V \):
\[ V = \frac{nRT}{P} \]

**Hint:** Isolate \( V \) by dividing both sides of the ideal gas equation by \( P \).

3. **Define Density:**
The density \( \rho \) of a substance is defined as its mass \( M \) divided by its volume \( V \):
\[ \rho = \frac{M}{V} \]

**Hint:** Density is a measure of how much mass is contained in a given volume.

4. **Substitute Volume into the Density Equation:**
Now, substitute the expression for volume \( V \) from step 2 into the density equation:
\[ \rho = \frac{M}{\frac{nRT}{P}} = \frac{MP}{nRT} \]

**Hint:** When substituting, remember to multiply by the reciprocal of the volume expression.

5. **Express Density in Terms of Molar Mass:**
If we consider \( M \) as the molar mass of the gas, we can express the number of moles \( n \) in terms of mass and molar mass:
\[ n = \frac{M}{M_m} \]
where \( M_m \) is the molar mass. Substituting this into the density equation gives:
\[ \rho = \frac{MP}{\left(\frac{M}{M_m}\right)RT} = \frac{PM_m}{RT} \]

**Hint:** Molar mass is the mass of one mole of a substance, and it helps relate mass to moles.

6. **Final Density Equation:**
Thus, the equation for the density of an ideal gas can be expressed as:
\[ \rho = \frac{PM}{RT} \]

**Hint:** This final equation shows how density is related to pressure, molar mass, and temperature.

### Summary:
The equation for the density of an ideal gas is:
\[ \rho = \frac{PM}{RT} \]