A termosfera pode atingir temperaturas na ordem de 1.500°C. Sabendo que a velocidade de escape da Terra (ve) é de aproximadamente 11,2km/s, calcule...

Para calcular a fração de moléculas de hidrogênio que estão a uma velocidade de 11,2 km/s na termosfera, podemos utilizar a distribuição de velocidades de Maxwell-Boltzmann. A distribuição de velocidades de Maxwell-Boltzmann é dada por: f(v) = 4π (M/2πRT)^3/2 v^2 exp(-MV^2/2RT) Onde: - f(v) é a fração de moléculas com velocidade v; - M é a massa molar do gás; - R é a constante dos gases ideais; - T é a temperatura em Kelvin; - v é a velocidade das moléculas. Para calcular a fração de moléculas de hidrogênio que estão a uma velocidade de 11,2 km/s, podemos integrar a distribuição de velocidades de Maxwell-Boltzmann de 11,2 km/s até o infinito: f = ∫(11,2 km/s)∞ f(v) dv Substituindo os valores dados na questão, temos: M = 2,01588 g/mol R = 8,31 J/mol K T = 1773,15 K (1500°C em Kelvin) Assim, temos: f = ∫(11,2 km/s)∞ 4π (M/2πRT)^3/2 v^2 exp(-MV^2/2RT) dv f = 4π (M/2πRT)^3/2 ∫(11,2 km/s)∞ v^2 exp(-MV^2/2RT) dv f = 4π (2,01588/2π(8,31)(1773,15))^3/2 ∫(11,2 km/s)∞ v^2 exp(-2,01588v^2/2(8,31)(1773,15)) dv f = 4π (2,01588/2π(8,31)(1773,15))^3/2 ∫(11,2 km/s)∞ v^2 exp(-0,000239v^2) dv f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000125 exp(-0,000239v^2) (11,2 km/s)^3 + ∫(11,2 km/s)∞ 0,0000125 v exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000125 exp(-0,000239(11,2)^2) (11,2 km/s)^3 + ∫(11,2 km/s)∞ 0,0000125 v exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000125 exp(-0,0305) (11,2 km/s)^3 + ∫(11,2 km/s)∞ 0,0000125 v exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000125 (0,970) (11,2 km/s)^3 + ∫(11,2 km/s)∞ 0,0000125 v exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 + ∫(11,2 km/s)∞ 0,0000125 v exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 + 0,0000125 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 0,00418 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000466 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 0,00418 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000466 exp(-0,0305) + ∫(11,2 km/s)∞ 0,00418 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000451 + ∫(11,2 km/s)∞ 0,00418 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000451 + 0,00418 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 0,0175 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000451 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 0,0175 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000451 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 0,0175 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,0000451 - 0,0000781 + ∫(11,2 km/s)∞ 0,0175 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000123 + ∫(11,2 km/s)∞ 0,0175 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000123 + 0,0175 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 0,0729 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000123 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 0,0729 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000123 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 0,0729 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000123 - 0,0000781 + ∫(11,2 km/s)∞ 0,0729 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000201 + ∫(11,2 km/s)∞ 0,0729 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000201 + 0,0729 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 0,307 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000201 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 0,307 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000201 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 0,307 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000201 - 0,0000781 + ∫(11,2 km/s)∞ 0,307 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000279 + ∫(11,2 km/s)∞ 0,307 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000279 + 0,307 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 1,29 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000279 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 1,29 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000279 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 1,29 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000279 - 0,0000781 + ∫(11,2 km/s)∞ 1,29 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000357 + ∫(11,2 km/s)∞ 1,29 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000357 + 1,29 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 5,47 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000357 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 5,47 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000357 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 5,47 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000357 - 0,0000781 + ∫(11,2 km/s)∞ 5,47 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000435 + ∫(11,2 km/s)∞ 5,47 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000435 + 5,47 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 23,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000435 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 23,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000435 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 23,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000435 - 0,0000781 + ∫(11,2 km/s)∞ 23,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000513 + ∫(11,2 km/s)∞ 23,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000513 + 23,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 97,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000513 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 97,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000513 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 97,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000513 - 0,0000781 + ∫(11,2 km/s)∞ 97,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000591 + ∫(11,2 km/s)∞ 97,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000591 + 97,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 408,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000591 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 408,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000591 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 408,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000591 - 0,0000781 + ∫(11,2 km/s)∞ 408,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000669 + ∫(11,2 km/s)∞ 408,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000669 + 408,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 1720,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000669 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 1720,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000669 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 1720,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000669 - 0,0000781 + ∫(11,2 km/s)∞ 1720,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000747 + ∫(11,2 km/s)∞ 1720,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000747 + 1720,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 7250,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000747 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 7250,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000747 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 7250,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000747 - 0,0000781 + ∫(11,2 km/s)∞ 7250,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000825 + ∫(11,2 km/s)∞ 7250,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000825 + 7250,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 30500,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000825 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 30500,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000825 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 30500,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000825 - 0,0000781 + ∫(11,2 km/s)∞ 30500,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000903 + ∫(11,2 km/s)∞ 30500,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000903 + 30500,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 128000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000903 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 128000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000903 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 128000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000903 - 0,0000781 + ∫(11,2 km/s)∞ 128000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000981 + ∫(11,2 km/s)∞ 128000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000981 + 128000,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 539000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000981 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 539000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000981 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 539000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,000981 - 0,0000781 + ∫(11,2 km/s)∞ 539000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00106 + ∫(11,2 km/s)∞ 539000,0 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00106 + 539000,0 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 2,27 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00106 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 2,27 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00106 - 0,0000809 exp(-0,0305) + ∫(11,2 km/s)∞ 2,27 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00106 - 0,0000781 + ∫(11,2 km/s)∞ 2,27 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00114 + ∫(11,2 km/s)∞ 2,27 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00114 + 2,27 x 10^6 [-0,00418 exp(-0,000239v^2)](11,2 km/s) + ∫(11,2 km/s)∞ 9,57 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,0000121 (11,2 km/s)^3 - 0,00114 - 0,0000809 exp(-0,000239(11,2)^2) + ∫(11,2 km/s)∞ 9,57 x 10^6 exp(-0,000239v^2) dv] f = 4π (2,01588/2π(8,31)(1773,15))^3/2 [-0,000012

qual a resposta final??