Edit to “Statement”

JMMA2006 edited Jun 24, 2026, 10:46 AM

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revision #19158 parent #19157 ← older

en/5.6.3.md +1 −1

		@@ -1,6 +1,6 @@
			### 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
			the room changed?


			### 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, 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 21

	@@ -1,6 +1,6 @@
	### 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

	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, but 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 21