A gas mixture contains 2 moles of helium (a monatomic gas) and N moles of hydrogen (a diatomic gas). The graph of the total internal energy U of the mixture versus its absolute temperature T is a straight line passing through the origin. If the slope of this graph is 23R , where R is the universal gas constant, the value of N is:

Step-by-step solution

The total internal energy U of a gas mixture is the sum of the internal energies of its components: U = n₁ f₁ 2 R T + n₂ f₂ 2 R T The slope of the U versus T graph is dU dT , which represents the total heat capacity at constant volume: Slope = dU dT = n₁ f₁ 2 R + n₂ f₂ 2 R For helium (monatomic), n₁ = 2 and f₁ = 3 . For hydrogen (diatomic), n₂ = N and f₂ = 5 . Substitute the given values into the slope equation: 23R = 2 ( 3 2 ) R + N ( 5 2 ) R 23R = 3R + 2.5 N R Dividing by R and solving for N : 23 = 3 + 2.5 N 2.5