We have an ideal gas problem with given pressure, temperature, and density. We need to analyze this gas step by step to find its molar mass, molecular properties, and behavior under different conditions.
To find the molar mass, we use the ideal gas law. Starting with PV equals nRT, we can derive that pressure equals density times R times T divided by molar mass. Rearranging gives us molar mass equals density times R times T divided by pressure. Substituting our values, we get 29.9 grams per mole, which corresponds to air.
Now we calculate the molecular properties. The average translational kinetic energy is three-halves k T, which equals 6.21 times 10 to the minus 21 joules. The root mean square speed is calculated using the formula involving R, T, and molar mass, giving us 503 meters per second for air molecules at this temperature.
For the final part, we analyze the new conditions where molecular density doubles and temperature increases to 127 degrees Celsius or 400 Kelvin. Using the relationship P equals n k T, the pressure ratio equals the product of density ratio and temperature ratio. This gives us 2 times 400 over 300, which equals 8 thirds. Therefore, the new pressure is 2.67 times 10 to the 5 pascals.