<Dd> v r m s = ∫ 0 ∞ v 2 p (v) d v (\ displaystyle v_ (\ mathrm (rms)) = (\ sqrt (\ int _ (0) ^ (\ infty) v ^ (2) \ p (v) dv)) \, \!) <Dl> <Dd> = ∫ 0 ∞ 4 π (m 2 π k T) 3 2 v 4 e − v 2 m 2 k T d v (\ displaystyle = (\ sqrt (\ int _ (0) ^ (\ infty) 4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) v ^ (4) \ e ^ (- (\ frac (v ^ (2) m) (2kT))) dv)) \, \!) </Dd> <Dd> = 4 π (m 2 π k T) 3 2 3 8 π 1 2 (2 k T m) 5 2 (\ displaystyle = (\ sqrt (4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) (\ frac (3) (8)) \ pi ^ (\ frac (1) (2)) \ left ((\ frac (2kT) (m)) \ right) ^ (\ frac (5) (2)))) \, \!) </Dd> <Dd> = 3 k T m (\ displaystyle = (\ sqrt (\ frac (3kT) (m)))) </Dd> </Dl> </Dd> <Dl> <Dd> = ∫ 0 ∞ 4 π (m 2 π k T) 3 2 v 4 e − v 2 m 2 k T d v (\ displaystyle = (\ sqrt (\ int _ (0) ^ (\ infty) 4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) v ^ (4) \ e ^ (- (\ frac (v ^ (2) m) (2kT))) dv)) \, \!) </Dd> <Dd> = 4 π (m 2 π k T) 3 2 3 8 π 1 2 (2 k T m) 5 2 (\ displaystyle = (\ sqrt (4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) (\ frac (3) (8)) \ pi ^ (\ frac (1) (2)) \ left ((\ frac (2kT) (m)) \ right) ^ (\ frac (5) (2)))) \, \!) </Dd> <Dd> = 3 k T m (\ displaystyle = (\ sqrt (\ frac (3kT) (m)))) </Dd> </Dl> <Dd> = ∫ 0 ∞ 4 π (m 2 π k T) 3 2 v 4 e − v 2 m 2 k T d v (\ displaystyle = (\ sqrt (\ int _ (0) ^ (\ infty) 4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) v ^ (4) \ e ^ (- (\ frac (v ^ (2) m) (2kT))) dv)) \, \!) </Dd> <Dd> = 4 π (m 2 π k T) 3 2 3 8 π 1 2 (2 k T m) 5 2 (\ displaystyle = (\ sqrt (4 \ pi \ left ((\ frac (m) (2 \ pi kT)) \ right) ^ (\ frac (3) (2)) (\ frac (3) (8)) \ pi ^ (\ frac (1) (2)) \ left ((\ frac (2kT) (m)) \ right) ^ (\ frac (5) (2)))) \, \!) </Dd>

Calculate the root mean square (rms) average speed