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

Answer:

a

$T_2 = 276.1 \ K$

b

  $V_2 = 2.13 *10^{3} \ m^3$

c

   $\Delta U = -1.25 *10^{7} \ J$

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

      $P_1 V_1 ^{\gamma }= P_2 V_2 ^{\gamma }$

=>    $V_2 ^ {\gamma } = \frac{ V_1 ^{\gamma } * P_1 }{P_2}$

Generally $\gamma$ is a constant with value $\gamma =\frac{5}{3}$ for an ideal  gas

So  

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

        $V_2 = (\sqrt[5]{103.14641852} )^3$

=>      $V_2 = 2.13 *10^{3} \ m^3$

Generally the adiabatic process can also be mathematically represented as

     $T_1 V_1 ^{\gamma -1 } = T_2 V_2^{\gamma -1 }$

=>  $T_2 = 288 * [\frac{2 * 10^{3}}{ 2.13 *10^{3}} ]^{ \frac{5}{3} -1 }$

=>    $T_2 = 276.1 \ K$

Generally the ideal gas equation is mathematically represented as

        $P_1 V_1 = nRT_1$

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

     $n = \frac{P_1 V_1 }{RT _1}$

=>  $n = \frac{1.0 *10^{5} * 2.0 *10^{3}}{8.314 * 288$

=>  $n = 84362 \ mol$

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  $C_v = 12.47 J/mol \cdot K$

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

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

          $\Delta U = -1.25 *10^{7} \ J$