A mass of \( 0,3 \mathrm{~kg} \) of a certain gas, of which the specific heat capacity at constant volume is \( 740 \mathrm{~J} / \mathrm{kgK} \), fllls a volume of 0,16 \( m^{\circ} \) at a pressure of 110 kPa and a temperature of \( 19^{\circ} \mathrm{C} \), calculate: a) the gas constant for this gas h) the specific heat canacity at a constant pressure for this gas

To find the gas constant \( R \) for the gas, we can use the ideal gas law, which states: \[ PV = nRT \] Where: - \( P \) is the pressure in pascals (Pa), - \( V \) is the volume in cubic meters (m³), - \( n \) is the number of moles, - \( R \) is the specific gas constant, - \( T \) is the temperature in kelvins (K). First, convert the pressure from kPa to Pa: \[ P = 110 \, \text{kPa} = 110,000 \, \text{Pa} \] Next, convert the temperature from Celsius to Kelvin: \[ T = 19^\circ C + 273.15 = 292.15 \, \text{K} \] Now, we convert the mass of gas to moles \( n \): \[ n = \frac{m}{M} \] However, we do not yet have the molar mass \( M \). Instead, we can express the gas constant \( R \) in terms of the specific heat capacities: Using the relation between specific heat capacity at constant volume (\( C_v \)) and the gas constant (\( R \)), we have: \[ C_p = C_v + R \] where \( C_p \) is the specific heat capacity at constant pressure. We can find \( R \) by determining \( C_p \) using the relation: Since \( C_v = 740 \, \text{J/kgK} \): Let's assume for a diatomic gas, \( R = C_p - C_v \). Using the relation for a diatomic gas, we have: \[ C_p \approx \frac{5}{2} R \] Thus, \[ R = C_v \cdot \frac{2}{3} \approx 740 \cdot \frac{2}{3} \approx 493.33 \, \text{J/(kg K)} \] Now, plugging \( R \) back to find \( C_p \): \[ C_p = C_v + R = 740 + 493.33 \approx 1233.33 \, \text{J/(kg K)} \] The gas constant \( R \approx 493.33 \, \text{J/(kg K)} \) and the specific heat capacity at constant pressure \( C_p \approx 1233.33 \, \text{J/(kg K)} \).