Edit to “Solution”
en/5.6.3.md
+1 −1
| ### Statement | |||
| $5.6.3.$ The air in the room was heated from the temperature T_0 to T. At the same | |||
| time, the pressure did not change. Has the internal energy of the air inside | |||
| the room changed? | |||
| ### Solution | |||
| @@ -10,7 +10,7 @@Solution | |||
| Assuming we're working with ideal gases\ | |||
| For an ideal gas, the internal energy only depends on the temperature\ | |||
| $U=nC_v T$ , where n is the number of moles and $C_v$ is the molar heat capacity at constant volume\ | |||
| − | Careful here, we could think that obviously if the temperature increases, the internal energy increases too, let's think this\ | ||
| + | Careful here, we could think that obviously if the temperature increases, the internal energy increases too, but let's think this\ | ||
| We have\ | |||
| $PV=nRT=const$, because P and V are both constant\ | |||
| but this is only possible if n changed (air can escape), because if n is constant, the product nRT would vary with T, contradicting that PV is constant\ | |||
| So, from $nRT=const$ , we get\ | |||
| $n_0 T_0=nT$\ | |||
| and substituting this into the expression for the final energy\ | |||
| $U=nC_v T$\ | |||
| $U=n_0 C_v T_0$ , which is exactly the initial energy $U_0$\ | |||
| so $U_0=U$\ | |||
| the internal energy did not change | |||
| #### Answer | |||
| The internal energy did not change | |||
| unchanged lines 11 | |||
| ### Statement | ### Statement | ||
| $5.6.3.$ The air in the room was heated from the temperature T_0 to T. At the same | $5.6.3.$ The air in the room was heated from the temperature T_0 to T. At the same | ||
| time, the pressure did not change. Has the internal energy of the air inside | time, the pressure did not change. Has the internal energy of the air inside | ||
| the room changed? | the room changed? | ||
| ### Solution | ### Solution | ||
| @@ -10,7 +10,7 @@Solution | |||
| Assuming we're working with ideal gases\ | Assuming we're working with ideal gases\ | ||
| For an ideal gas, the internal energy only depends on the temperature\ | For an ideal gas, the internal energy only depends on the temperature\ | ||
| $U=nC_v T$ , where n is the number of moles and $C_v$ is the molar heat capacity at constant volume\ | $U=nC_v T$ , where n is the number of moles and $C_v$ is the molar heat capacity at constant volume\ | ||
| Careful here, we could think that obviously if the temperature increases, the internal energy increases too, let's think this\ | Careful here, we could think that obviously if the temperature increases, the internal energy increases too, but let's think this\ | ||
| We have\ | We have\ | ||
| $PV=nRT=const$, because P and V are both constant\ | $PV=nRT=const$, because P and V are both constant\ | ||
| but this is only possible if n changed (air can escape), because if n is constant, the product nRT would vary with T, contradicting that PV is constant\ | but this is only possible if n changed (air can escape), because if n is constant, the product nRT would vary with T, contradicting that PV is constant\ | ||
| So, from $nRT=const$ , we get\ | So, from $nRT=const$ , we get\ | ||
| $n_0 T_0=nT$\ | $n_0 T_0=nT$\ | ||
| and substituting this into the expression for the final energy\ | and substituting this into the expression for the final energy\ | ||
| $U=nC_v T$\ | $U=nC_v T$\ | ||
| $U=n_0 C_v T_0$ , which is exactly the initial energy $U_0$\ | $U=n_0 C_v T_0$ , which is exactly the initial energy $U_0$\ | ||
| so $U_0=U$\ | so $U_0=U$\ | ||
| the internal energy did not change | the internal energy did not change | ||
| #### Answer | #### Answer | ||
| The internal energy did not change | The internal energy did not change | ||
| unchanged lines 11 | |||