Both expressions are valid for calculating the RMS speed of a monoatomic ideal gas.

The RMS speed increases with temperature as v_rms = sqrt(3RT/M) shows that it is directly proportional to the square root of temperature T.

According to Boyle's law, for a given mass of gas at constant temperature, the pressure is inversely proportional to the volume. Halving the volume will double the pressure.

The equation of state for an ideal gas is given by PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

The ideal gas law is given by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

The ideal gas law is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

At constant temperature, Boyle's law states that the product of pressure and volume (PV) is constant, hence P1V1 = P2V2.

At STP (Standard Temperature and Pressure), 1 mole of an ideal gas occupies 22.4 L at a pressure of 1 atm.

Using the Ideal Gas Law, P = nRT/V = (2 moles * 0.0821 L·atm/(K·mol) * 300 K) / 10 L = 4.926 atm.

Using Boyle's Law (P1V1 = P2V2), if P1 = 1 atm, V1 = 10 L, and P2 = 2 atm, then V2 = (P1V1)/P2 = (1 atm * 10 L) / 2 atm = 5 L.

Using Boyle's Law (P1V1 = P2V2), if the pressure doubles from 1 atm to 2 atm, the volume will halve from 10 L to 5 L.

RMS speed is inversely proportional to the square root of molar mass, so if M is doubled, v_rms decreases by sqrt(2).

The RMS speed is inversely proportional to the square root of the molar mass. If M is doubled, v_rms decreases by sqrt(2).

If the molar mass is halved, the RMS speed increases by a factor of sqrt(2) because RMS speed is inversely proportional to the square root of molar mass.

Using v_rms = sqrt(3RT/M), we can rearrange to find T = (v_rms^2 * M) / (3R) = 300 K.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T = (v_rms^2 * M)/(3R). Plugging in the values gives T = 600 K.

The RMS speed is proportional to the square root of the temperature. Therefore, at 600 K, the RMS speed will be 300 * sqrt(2) m/s.

The RMS speed is proportional to the square root of the temperature. Therefore, at 200 K, the RMS speed will be 300 * sqrt(200/400) = 150 m/s.

RMS speed is proportional to the square root of temperature. v_rms at 800 K = 300 * sqrt(800/400) = 300 * sqrt(2) = 600 m/s.

The RMS speed is proportional to the square root of the temperature. If the temperature is doubled, the RMS speed increases by a factor of sqrt(2). Therefore, the new RMS speed will be 300 * sqrt(2), which is approximately 600 m/s.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T: T = (M * v_rms^2) / (3R). Substituting M = 0.016 kg/mol and v_rms = 400 m/s gives T = 400 K.

The RMS speed increases with the square root of the temperature. Therefore, at 600 K, the RMS speed will be 400 * sqrt(2), which is approximately 800 m/s.

Kinetic energy per molecule = (1/2)mv^2. Using v = 400 m/s and m = M/N_A, we find KE = 0.5 mJ.

The average speed is related to RMS speed by the relation v_avg = (2/3) * v_rms. Thus, v_avg = (2/3) * 400 = 267 m/s.

The average speed of gas molecules is approximately 0.8 times the RMS speed, so average speed = 0.8 * 400 m/s = 320 m/s.

The speed at 1/2 of the RMS speed is simply 500 m/s / 2 = 250 m/s.

The average speed of gas molecules is related to the RMS speed by the relation v_avg = (v_rms * sqrt(8/3)). Therefore, the average speed is approximately 400 m/s.

The RMS speed is an average measure; individual molecular speeds will vary around this value.

Using v_rms = sqrt(3RT/M), we rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T approximately 300 K.

According to Gay-Lussac's Law, if the temperature of a gas is increased while keeping the volume constant, the pressure will also increase proportionally, thus it doubles.