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en/5.6.3.md
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| + | ### Statement | ||
| + | |||
| + | $5.6.3.$ [Insert the problem statement] | ||
| + | |||
| + | ### 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\ | ||
| + | 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 | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $5.6.3.$ [Insert the problem statement] | |||
| ### 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\ | |||
| 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 | |||
| [Insert a concise answer or boxed result] | |||