The weight of a gas is given as 17.658 N/m³ at 30°C and at an absolute pressure of 29.43 N/cm². Determine the gas constant and also the density of the gas.

Gas Constant and Density Calculation

Problem Statement

The specific weight of a gas is given as 17.658 N/m³ at 30°C and at an absolute pressure of 29.43 N/cm². Determine the gas constant and also the density of the gas.

Given Data

Specific Weight, $\gamma = 17.658 \, \text{N/m}^3$
Absolute Pressure, $P = 29.43 \, \text{N/cm}^2$
Temperature, $t = 30^\circ\text{C}$

Solution

1. Convert Units to SI

We convert the given pressure and temperature to their standard SI units (N/m² and Kelvin) for use in the formulas.

Pressure Conversion:

$$ P = 29.43 \, \frac{\text{N}}{\text{cm}^2} \times \frac{10000 \, \text{cm}^2}{1 \, \text{m}^2} $$ $$ P = 294300 \, \text{N/m}^2 $$

Temperature Conversion:

$$ T = 30^\circ\text{C} + 273.15 $$ $$ T = 303.15 \, \text{K} $$

2. Calculate the Density (ρ)

Density ($\rho$) can be found from the specific weight ($\gamma$) using the acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$.

$$ \gamma = \rho \times g $$ $$ \rho = \frac{\gamma}{g} $$ $$ \rho = \frac{17.658 \, \text{N/m}^3}{9.81 \, \text{m/s}^2} $$ $$ \rho = 1.8 \, \text{kg/m}^3 $$

3. Calculate the Gas Constant (R)

The gas constant ($R$) can be determined using the Ideal Gas Law, expressed in terms of density: $P = \rho R T$.

$$ R = \frac{P}{\rho T} $$ $$ R = \frac{294300 \, \text{N/m}^2}{1.8 \, \text{kg/m}^3 \times 303.15 \, \text{K}} $$ $$ R \approx 538.35 \, \frac{\text{N·m}}{\text{kg·K}} \text{ or } \, \text{J/kg·K} $$

Final Results:

Density of the gas, $ \rho = 1.8 \, \text{kg/m}^3 $

Gas Constant, $ R \approx 538.35 \, \text{J/kg·K} $

Explanation of Key Concepts

Specific Weight ($\gamma$): The weight of a substance per unit volume. It is dependent on gravity ($\gamma = \rho g$).
Density ($\rho$): The mass of a substance per unit volume. It is an intrinsic property of the material.
Ideal Gas Law: A fundamental equation of state that relates the pressure, temperature, and volume of a gas. In the form $P = \rho R T$, it connects these properties with the gas's density and its specific gas constant.
Gas Constant (R): A characteristic constant for a particular gas. It is different for every gas (e.g., air, helium, argon) and is a key part of its thermodynamic identity.

Physical Meaning

The calculated density of 1.8 kg/m³ tells us the mass of the gas contained in one cubic meter. For comparison, the density of air at a similar temperature and standard pressure is about 1.16 kg/m³, so this gas is denser than air.

The gas constant of 538.35 J/kg·K is a unique property of this specific gas. This value allows engineers and scientists to predict how the gas's pressure, volume, and temperature will change under different conditions. For example, the gas constant for air is about 287 J/kg·K. The significantly different value indicates this is not air, but another type of gas (it is close to the gas constant for methane).