南开大学物理化学2-1第二章——热力学第一定律公式合集
第二章
1. 热力学的一些基本概念
- \(\alpha(T,p)\equiv\frac{1}{V}(\frac{\partial{V}}{\partial{T}})_{p,n}=\frac{1}{V_m}(\frac{\partial{V_m}}{\partial{T}})_p\)
- \(\kappa(T,p)\equiv-\frac{1}{V}(\frac{\partial{V}}{\partial{p}})_{T,n}=-\frac{1}{V_m}(\frac{\partial{V_m}}{\partial{p}})_T\)
- \((\frac{\partial{p}}{\partial{T}})_{V_m}=\frac{\alpha}{\kappa}\)
- \(W=-\sum\limits_ip_{ex,i}\Delta{V_i}\) 不连续有限过程（\(p_ex\)为外压）
- \(W=-\int_1^2p_{ex}dV\) 连续过程
- \(W=0\) 真空膨胀
- 可逆膨胀
- \(W=-nRT\ln{\frac{V_2}{V_1}}\) 理想气体恒温
- \(W=-[\frac{an^2}{V}+nRT\ln(V-nb)]|_{V_1}^{V_2}\) 真实气体恒温
- \(\oint{dU}=0, \oint{dH}=0, \oint{dS}=0, \oint{dA}=0, \oint{dG}=0\)
- \(dU=(\frac{\partial{U}}{\partial{T}})_VdT+(\frac{\partial{U}}{\partial{V}})_TdV=C_VdT+(T\frac{\alpha}{\kappa}-p)dV\) 吉布斯方程适用条件（见第三章公式合集）
2. 热力学第一定律
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\(U=Q+W\)
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\(dU=\delta{Q}+\delta{W}\)
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\(H=U+pV\)
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\(dH=dU+d(pV)=dU+pdV+Vdp\)
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\(\Delta{H}=\Delta{U}+\Delta{pV}=\Delta{U}+(p_2V_2-p_1V_1)\)
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\(dH=(\frac{\partial{H}}{\partial{T}})_pdT+(\frac{\partial{H}}{\partial{p}})_Tdp\) 组成恒定封闭体系
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\(dH=C_pdT+(V-TV\alpha)dp\) 吉布斯方程适用条件
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\(dH=\delta{Q_p},\ \Delta{H}=Q_p\) 恒压只做体积功的封闭体系
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\(dU=\delta{Q_V},\ \Delta{U}=Q_V\) 恒容封闭体系
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恒容热容
- \(C_V\equiv\frac{\delta{Q_V}}{dT}\equiv(\frac{\partial{U}}{\partial{T}})_V\) 恒容封闭体系
- \(C_{V,m}=\frac{C_V}{n}\) 纯物质(若考虑平均热容则可是用于组成恒定体系
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恒压热容
- \(C_p\equiv\frac{\delta{Q_p}}{dT}\equiv(\frac{\partial{H}}{\partial{T}})_p\) 恒容封闭体系
- \(C_{p,m}=\frac{C_p}{n}\) 纯物质(若考虑平均热容则可是用于组成恒定体系
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\(C_p-C_V=[p+(\frac{\partial{U}}{\partial{V}})_T](\frac{\partial{V}}{\partial{T}})_p\) 组成恒定封闭体系
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\(Q_p=\int_1^2nC_{p,m}dT\) 无相变化只做体积功组成恒定的封闭恒压体系
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\(Q_V=\int_1^2nC_{V,m}dT\) 无相变化只做体积功组成恒定的封闭恒容体系
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\((\frac{\partial{U}}{\partial{V}})_T=0\) 理想气体
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\(\mu_J\equiv(\frac{\partial{T}}{\partial{V}})_U\)
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\(\mu_J=0\) 理想气体
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\((\frac{\partial{U}}{\partial{V}})_T=-C_V\mu_J\)
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\((\frac{\partial{H}}{\partial{p}})_T=(\frac{\partial{H}}{\partial{V}})_T=0\) 理想气体
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理想气体的恒压热容与恒容热容
- \(C_p-C_V=nR\)
- \(C_{p,m}-C_{V,m}=R\)
- \(C_V=\frac{3}{2}{R},\ C_p=\frac{5}{2}R\) 单原子分子（3个自由度，具体推导见第五章）
- \(C_V=\frac{5}{2}{R},\ C_p=\frac{7}{2}R\) 双原子分子（5个自由度，具体推导见第五章）
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\((\frac{\partial{H}}{\partial{p}})_T=-C_p\mu_{J-T}\)
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\(\mu_{J-T}\equiv(\frac{\partial{T}}{\partial{p}})_H\)
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绝热条件理想气体做功
- \(\gamma=\frac{C_{p,m}}{C_{V,m}}\) 无需满足理想气体
- \(\gamma=1+\frac{R}{C_{V,m}}\)
- \(W=\frac{nR(T_2-T_1)}{\gamma-1}=\frac{p_2V_2-p_1V_1}{\gamma-1}\) 是否可逆都满足
- \(W=-\int_1^2p_{ex}dV=C_V(T_2-T_1)\) 是否可逆都满足
- 绝热可逆
- \(T_1V_1^{\gamma-1}=T_2V_2^{\gamma-1}\)
- \(p_1V_1^{\gamma}=p_2V_2^{\gamma}\)
- \(\frac{T_1^\gamma}{p_1^{\gamma-1}}=\frac{T_2^\gamma}{p_2^{\gamma-1}}\)
3. 相变焓
- \(\Delta_{vap}H_m^\Theta\equiv{H_m^\Theta}(g)-H_m^\Theta(l)\)
- \(\Delta_{fus}H_m^\Theta\equiv{H_m^\Theta}(l)-H_m^\Theta(s)\)
- \(\Delta_{sub}H_m^\Theta\equiv{H_m^\Theta}(g)-H_m^\Theta(s)\)
- 以下方程均需满足可逆相变恒压只做体积功
- \(\Delta_{vap}H_m^\Theta=Q_p\)
- \(\Delta_{fus}H_m^\Theta=Q_p\)
- \(\Delta_{sub}H_m^\Theta=Q_p\)
4. 反应热力学量变
- \(dn_B=\nu{d\zeta}\)
- \(\Delta_rU_m=\frac{\Delta_rU}{\Delta\zeta},\ \Delta_rH_m=\frac{\Delta_rH}{\Delta\zeta}\)
- 理想气体，且忽略液体固体体积变化
- \(\Delta_rH=\Delta_rU+\Delta{n_g}RT\)
- \(\Delta_rH_m=\Delta_rU_m+\Delta{\nu_g}RT\)
- 规定
- \(H_m^\Theta(B,298.15K)\equiv\Delta_fH_m^\Theta(B,298.15K)\)
- \(H_m^\Theta(稳定单质,298.15K)\equiv0\)
- \(\Delta_rH_m^\Theta(298.15K)=-\sum\limits_B\nu_B\Delta_cH_m^\Theta(B,298.15K)\)
- \(\Delta_rH_m^\Theta(298.15K)=\sum\limits_B\nu_B\Delta_fH_m^\Theta(B,298.15K)\)
- \(\Delta_rH_m^\Theta(298.15K)=-\sum\limits_B\nu_B\Delta_{at}H_m^\Theta(B,298.15K,g)\) 只包含气态物质
- \(\Delta_rH_m^\Theta(T_2)-\Delta_rH_m^\Theta(T_1)=\int_{T_1}^{T_2}\Delta{C_p^\Theta}dT\) 无相变封闭体系
- \(\frac{d\Delta_rH_m^\Theta}{dT}=\sum\limits_B\nu_BC_{p,m}^\Theta(B)=\Delta{C_{p,m}^\Theta}\) 无相变封闭体系
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