Question

A large research balloon containing 2.00 × 10^3 m^3 of helium gas at 1.00 atm and a temperature of 15.0°C rises rapidly from ground level to an altitude at which the atmospheric pressure is only 0.900 atm. Assume the helium behaves like an ideal gas and the balloon’s ascent is too rapid to permit much heat exchange with the surrounding air.

Required:
a. Calculate the volume of the gas at the higher altitude.
b. Calculate the temperature of the gas at the higher altitude.
c. What is the change in internal energy of the helium as the balloon rises to the higher altitude?

Answer 1

a

`T_2 =  276.1 \ K`

b

`V_2  = 2.13 *10^(3) \ m^3`

c

`\Delta U =  -1.25 *10^(7) \  J`

Step-by-step explanation:

From the question we are told that

The volume of the balloon is
`V  =  2.00 * 10^3 \ m^3`

The pressure of helium is
`P_1 =   1.00 \ atm=  1.0 *10^(5) \  Pa`

The initial temperature is
`T_1   =  15.0^oC =  288 \  K`

The pressure of atmosphere is
`P_a  =  0.900 \ atm`

Generally the equation representing the adiabatic process is mathematically represented as

`P_1 V_1 ^(\gamma )=  P_2 V_2 ^(\gamma )`

=>
`V_2 ^ {\gamma } =  ( V_1 ^(\gamma ) *  P_1 )/(P_2)`

Generally
`\gamma` is a constant with value
`\gamma  =(5)/(3)` for an ideal gas

So

`V_2 ^ {(5)/(3) } =  \frac{ ( 2.0 *10^(3)) ^{ (5)/(3)  } *   1.00 }{0.900}`

`V_2  =  (\sqrt[5]{103.14641852} )^3`

=>
`V_2  = 2.13 *10^(3) \ m^3`

Generally the adiabatic process can also be mathematically represented as

`T_1 V_1 ^(\gamma -1 ) = T_2 V_2^(\gamma -1 )`

=>
`T_2  =  288 *  [(2 *  10^(3))/( 2.13 *10^(3)) ]^{ (5)/(3) -1 }`

=>
`T_2 =  276.1 \ K`

Generally the ideal gas equation is mathematically represented as

`P_1 V_1 = nRT_1`

Here R is the gas constant with value
`R  =  8.314\  J  /mol \cdot  K`

`n  =  (P_1 V_1 )/(RT _1)`

=>

=>
`n  =  84362 \  mol`

Generally change in internal energy i mathematically represented

`\Delta U =  n C_v  \Delta  T`

Here
`C_v` is the specific heats of gas at constant volume and the value is
`C_v  =  12.47 J/mol \cdot  K`

`\Delta U =  84362 *   12.47 * [T_2 - T_1 ]`

`\Delta U =  84362 *   12.47 * [276.1 - 288 ]`

`\Delta U =  -1.25 *10^(7) \  J`