Answer:
a
[tex]T_2 = 276.1 \ K[/tex]
b
[tex]V_2 = 2.13 *10^{3} \ m^3 [/tex]
c
[tex]\Delta U = -1.25 *10^{7} \ J [/tex]
Explanation:
From the question we are told that
The volume of the balloon is [tex]V = 2.00 * 10^3 \ m^3[/tex]
The pressure of helium is [tex]P_1 = 1.00 \ atm= 1.0 *10^{5} \ Pa [/tex]
The initial temperature is [tex]T_1 = 15.0^oC = 288 \ K [/tex]
The pressure of atmosphere is [tex]P_a = 0.900 \ atm[/tex]
Generally the equation representing the adiabatic process is mathematically represented as
[tex]P_1 V_1 ^{\gamma }= P_2 V_2 ^{\gamma }[/tex]
=> [tex]V_2 ^ {\gamma } = \frac{ V_1 ^{\gamma } * P_1 }{P_2}[/tex]
Generally [tex]\gamma[/tex] is a constant with value [tex]\gamma =\frac{5}{3}[/tex] for an ideal gas
So
[tex]V_2 ^ {\frac{5}{3} } = \frac{ ( 2.0 *10^{3}) ^{ \frac{5}{3} } * 1.00 }{0.900}[/tex]
[tex]V_2 = (\sqrt[5]{103.14641852} )^3[/tex]
=> [tex]V_2 = 2.13 *10^{3} \ m^3 [/tex]
Generally the adiabatic process can also be mathematically represented as
[tex]T_1 V_1 ^{\gamma -1 } = T_2 V_2^{\gamma -1 }[/tex]
=> [tex]T_2 = 288 * [\frac{2 * 10^{3}}{ 2.13 *10^{3}} ]^{ \frac{5}{3} -1 }[/tex]
=> [tex]T_2 = 276.1 \ K[/tex]
Generally the ideal gas equation is mathematically represented as
[tex]P_1 V_1 = nRT_1[/tex]
Here R is the gas constant with value [tex]R = 8.314\ J /mol \cdot K[/tex]
[tex]n = \frac{P_1 V_1 }{RT _1}[/tex]
=> [tex]n = \frac{1.0 *10^{5} * 2.0 *10^{3}}{8.314 * 288[/tex]
=> [tex]n = 84362 \ mol[/tex]
Generally change in internal energy i mathematically represented
[tex]\Delta U = n C_v \Delta T[/tex]
Here [tex]C_v[/tex] is the specific heats of gas at constant volume and the value is [tex]C_v = 12.47 J/mol \cdot K[/tex]
[tex]\Delta U = 84362 * 12.47 * [T_2 - T_1 ] [/tex]
[tex]\Delta U = 84362 * 12.47 * [276.1 - 288 ] [/tex]
[tex]\Delta U = -1.25 *10^{7} \ J [/tex]
