南开大学物理化学2-1第六章——液体混合物和溶液公式合集

第六章

---

Cotent

• 基本物理量定义
• 偏摩尔量相关
• 理想液体混合物
• 非理想液体混合物
• 电解质溶液
• 热力学函数规定值

基本物理量定义

• \(x_B=\frac{n_b}{\sum\limits_{B}{n_B}}\) (B为任一成分，在与A同时出现时专指溶质，后同)
• $m_B=\frac{n_B}{m_A}=\frac{n_B}{M_An_A} $ (A为溶剂）
• \(c_B=\frac{n_B}{V}\)
• \(x_B=\frac{m_BN_A}{1+\sum\limits_{B}{m_BN_A}}\) \(m_B=\frac{x_B}{M_A(1-\sum\limits_{C\neq{A}}x_C)}\)
• \(x_B=\frac{c_BM_A}{\rho+c_B(M_A-M_B)}\)   二组分体系
• \(W_B=\frac{m_B}{\sum\limits_Bm_B}\)

偏摩尔量相关

• \(X_B=(\frac{\partial{X}}{\partial{n_B}})_{T,p,n_{j\neq{B}}}\)
• \(\mu_B=G_B\)
• 集合公式
  • \(X=\sum\limits_Bn_BX_B\)
  • \(X_m=\sum\limits_{B=1}^{k}x_BX_B\)
• \(\mu_B=G_B=H_B-TS_B\)
• \((\frac{\partial{\mu_B}}{\partial{T}})_{p,n}=(\frac{\partial{G_B}}{\partial{T}})_{p,n}=-S_B\)
• \((\frac{\partial{\mu_B}}{\partial{p}})_{T,n}=(\frac{\partial{G_B}}{\partial{p}})_{T,n}=V_B\)
• \((\frac{\partial{\frac{G_B}{T}}}{\partial{T}})_{p,n}=(\frac{\partial{\frac{\mu_B}{T}}}{\partial{T}})_{p,n}=-\frac{H_B^2}{T^2}\)
• \(dG_B=d\mu_B=-S_BdT+V_Bdp\)
• \(\sum\limits_B n_B dX_B=(\frac{\partial{X}}{\partial{T}})_{p,n}dT+(\frac{\partial{X}}{\partial{p}})_{T,n}dp\)
  • 恒温恒压
    • \(\sum\limits_B n_B dX_B=0\) , \(\sum\limits_B x_B d{X_B}=0\)
    • 二组分均相体系
      • \(dX_A=-\frac{x_B}{x_A}dX_B\)
• 实验测定偏摩尔量方法
  • 截距法
    • \(X_m=(X_B-X_A)x_B+X_A\to\)
      以\(x_B\)为横轴，\(X_m\)为纵轴作图，与目标体系处做切线，交\(x_B=0\)于\((0,X_A)\)，交\(x_B=1\)于\((1,X_B)\)
  • 斜率法
    • \(X_B=(\frac{\partial{X}}{\partial{n_B}})_{T,p,n_{j\neq{B}}}\to\)
      固定别的组分、\(T\)、\(P\)不变，做\(X\)关于\(n_B\)的曲线，在目标体系处做切线求斜率即为目标偏摩尔量

理想液体混合物

• 拉乌尔定律
  • \(p_A=p_A^*x_A\) \((x_A\to1)\) 理想液体混合物的话，在\(0\to1\)均满足
• 亨利定律 \((x_B\to0)\)
  • \(p_B=K_{x,B}x_B\)
  • \(p_B=K_{m,B}m_B\)
  • \(p_B=K_{c,B}c_B\)
  • \(K_{x,B}=\frac{1}{M_A}K_{m,B}=\frac{\rho}{M_A}K_{c,B}\)
• 理想液体混合物
  • \(\mu_B(l,T,p,x_B)=\mu^*_B(l,T,p^*)+RT\ln{x_B}\approx\mu^*_B(l,T,p^{\theta})+RT\ln{x_B}=\mu^\theta_B(l,T)+RT\ln{x_B}\)
  • \(H_B=H^*_{m,B}\)
  • \(V_B=V^*_{m,B}\)
  • \(S_B=S^*_{m,B}\)
  • \(\Delta_{mix}V=\Delta_{mix}H=\Delta{mix}U=0\)
  • \(\Delta_{mix}S=-R\sum\limits_Bn_B\ln{x_B}\)
• \(\Delta_{mix}Z=\sum\limits_Bn_B(Z_B-Z_{m,B}^*)\)
• 理想稀溶液
  • \(\mu_B(l,T,p,x_B)=\mu^{\theta}_{B,x}(T)+RT\ln{x_B}=\mu_B^{\theta}(g,T)+RT\ln\frac{K_{x,B}}{p^{\theta}}+RT\ln{x_B}\)
  • \(\mu_B(l,T,p,c_B)=\mu^{\theta}_{B,c}(T)+RT\ln\frac{c_B}{c^\theta}\)
  • \(\mu_B(l,T,p,c_B)=\mu^{\theta}_{B,m}(T)+RT\ln\frac{c_B}{m^\theta}\)
  • \(\Delta_{mix}V=\sum\limits_{B\neq{A}}n_B(V_B-V^*_{m,B})\neq{0}\)
  • \(\Delta_{mix}H=\sum\limits_{B\neq{A}}n_B(H_B-H^*_{m,B})\neq{0}\)
  • \(\Delta_{mix}S=-n_AR\ln{x_A}+\sum\limits_{B\neq{A}}n_B(S_B^{\theta}-R\ln{x_B}-V^*_{m,B})\neq{0}\)
  • \(\Delta_{mix}G=n_ART\ln{x_A}+\sum\limits_{B\neq{A}}n_B(\mu_B^{\theta}+RT\ln{x_B}-\mu^*_{m,B})\neq{0}\)
• \(\Delta_{sol}H_m^I=\frac{\Delta_{mix}H}{n_B}=\frac{n_B}{n_A}(H_A-H_{m,A}^*)+(H_B-H_{m,B}^*)\)
• \(\Delta_{dil}H_m^I=\Delta_{sol}H_m^I(终)-\Delta_{sol}H_m^I(初)\)
• \(\Delta_{sol}H_{m,B}^D=(\frac{\partial{\Delta_{mix}H}}{\partial{n_B}})_{T,p,n_A}=H_B-H_{m,B}^*\)

非理想液体混合物

• \(a_B=exp\frac{\mu_B-\mu_B^{\theta}}{RT}\)
• \(\gamma_B=\frac{a_B}{x_B}\begin{cases}\gamma>1,蒸气压正偏差\\\gamma<1,蒸气压负偏差\end{cases}\)
• 转移性质
  • \(\Delta{G}_{T,p}=n_BRT\ln\frac{a_B(II)}{a_B(I)}\)
  • 理想液体混合物
    • \(\Delta{G}_{T,p}=n_BRT\ln\frac{x_B(II)}{x_B(I)}\)
    • \(\Delta{S}_{T,p}=-n_BR\ln\frac{x_B(II)}{x_B(I)}\) 注意：非理想液体混合物对于T求导不能直接去T，因为a可能是T的函数
• 超额热力学函数
  • \(X^E=\Delta_{mix}X-\Delta_{mix}X^{id}\)
  • \(G^E=\Delta_{mix}G-\Delta_{mix}G^{id}=RT\sum\limits_Bn_B\ln{\gamma_B}\)
  • \(S^E=\Delta_{mix}S-\Delta_{mix}S^{id}=-R\sum\limits_Bn_B\ln{\gamma_B}-RT\sum\limits_Bn_B(\frac{\partial\ln{\gamma_B}}{\partial{T}})_{p,n}\)
  • \(H^E=\Delta_{mix}H-\Delta_{mix}H^{id}=-RT^2\sum\limits_Bn_B(\frac{\partial\ln{\gamma_B}}{\partial{T}})_{p,n}\)
  • \(V^E=\Delta_{mix}V-\Delta_{mix}V^{id}=RT\sum\limits_Bn_B(\frac{\partial\ln{\gamma_B}}{\partial{p}})_{T,n}\)
  • \(G^E=H^E-TS^E\)
  • \((\frac{\partial{G^E}}{\partial{p}})_{T,n}=V^E\)
  • \((\frac{\partial{G^E}}{\partial{T}})_{p,n}=-S^E\)
• \(a_{B,m}=\gamma_{B,m}\frac{m_B}{m_B^{\theta}}\)
• \(a_{B,c}=\gamma_{B,c}\frac{c_B}{c_B^{\theta}}\)
• \(a_{B,x}=\gamma_{B,x}{x_B}\)
• \(\frac{\gamma_{B,m}}{\gamma_{B,x}}=\frac{1}{1+M_Am_B}\)
• \(\frac{a_{B,m}}{a_{B,x}}=\frac{1}{M_Am^{\theta}}\)
• 二组分体系
  • \(\Delta{p}=p_A^*x_B\)
  • \(M_B=\frac{W_BM_A}{W_A}\frac{p_A^*}{\Delta{p}}\)
• \(\ln{a_A}=\frac{\Delta_s^lH_{m,A}^*}{R}(\frac{1}{T_f^*}-\frac{1}{T_f})\)
• \(\Delta{T_f}=T_f^*-T_f=\frac{M_AR(T_f^*)^2m_B}{\Delta_s^lH_{m,A}^*}=\frac{R(T_f^*)^2x_B}{\Delta_s^lH_{m,A}^*}=K_fm_B\) 理想二组分稀溶液
• \(M_B=\frac{W_B}{W_Am_B}=\frac{K_fW_B}{W_A\Delta{T_f}}\) 理想二组分稀溶液
• \(\ln{a_A}=\frac{\Delta_l^gH_{m,A}^*}{R}(\frac{1}{T_b}-\frac{1}{T_b^*})\)
• \(\Delta{T_b}=T_b-T_b^*=\frac{M_AR(T_b^*)^2m_B}{\Delta_l^gH_{m,A}^*}=\frac{R(T_b^*)^2x_B}{\Delta_l^gH_{m,A}^*}=K_bm_B\) 理想二组分稀溶液
• \(\Pi=-(\frac{RT}{V^*_A})\ln{(\gamma_Ax_A)}\)
• \(\Pi=(\frac{RT}{V_A^*})x_B\) 理想二组分稀溶液
• \(d\ln\gamma_B=-\frac{x_A}{x_B}d\ln{\gamma_A}\) 二组分稀溶液

电解质溶液

• \(\mu_B=(\frac{\partial{G}}{\partial{n_B}})_{T,p,n_A}=\nu_+\mu_++\nu_-\mu_-\)
• \(\begin{align*}\mu_B&=\mu_B^{\theta}+RT\ln[(\gamma_+\frac{m_+}{m^{\theta}})^{\nu_+}(\gamma_-\frac{m_-}{m^{\theta}})^{\nu_-}]\\&=\mu_B^{\theta}+RT\ln[\gamma_{\pm}^{\nu}(\frac{m_\pm}{m^\theta})^\nu]\\&=\mu_B^\theta+RT\ln{a_{\pm}^\nu}\end{align*}\)
• \(\nu\equiv\nu_++\nu_-\ \ \ \ \nu_\pm^\nu=\nu_+^{\nu_+}+\nu_-^{\nu_-}\)
• \(\gamma_\pm^\nu=\gamma_+^{\nu_+}+\gamma_-^{\nu_-}\) \(m_\pm^\nu=m_+^{\nu_+}+m_-^{\nu_-}\) \(a_\pm^\nu=a_+^{\nu_+}+a_-^{\nu_-}\)
• \(I=\frac{1}{2}\sum\limits_jm_jz_j^2\)
• 25 °C, 101325Pa
  • \(\lg{\gamma_{\pm}}=-0.509z_+|z_-|\sqrt{I_m}\)
  • \(\lg{\gamma_{+}}=-0.509z_+^2\sqrt{I_m}\)
  • \(\lg{\gamma_{-}}=-0.509z_-^2\sqrt{I_m}\)
• 25 °C, 10⁵Pa
  • \(\lg{\gamma_{\pm}}=-0.5115z_+|z_-|\sqrt{I_m}\)
  • \(\lg{\gamma_{+}}=-0.5115z_+^2\sqrt{I_m}\)
  • \(\lg{\gamma_{-}}=-0.5115z_-^2\sqrt{I_m}\)

热力学函数规定值

• \(\Delta_fG_m^\theta(B,soln,T)=\Delta_fG_m^{\theta}(B,T)-RT\ln{\frac{\gamma_Bm_B^{sat}}{m^\theta}}\)
• 规定
  • \(\Delta_fG_m^\theta(H^+,aq,T)\equiv{0}\)
  • \(\Delta_fH_m^\theta(H^+,aq,T)\equiv{0}\)
  • \(\Delta_fS_m^\theta(H^+,aq,T)\equiv{0}\)
  • \(S_m^\theta(H^+,aq,T)\equiv{0}\)
  • \(C_{p,m}^\theta(H^+,aq,T)\equiv{0}\)
• \(\Delta_fG_m^\theta(B,soln,T)=\Delta_fG_m^{\theta}(B,T)-RT\ln({\frac{\gamma_Bm_B^{sat}}{m^\theta}})^\nu\)
• \(\Delta_fG_m^\theta(B,aq,T)=\nu_+[\mu_+-G_{e(+)}^\theta]+\nu_-[\mu_--G_{e(-)}^\theta]\)

标准态：

• 理想液体混合物：同温度、标准压力\(p^\theta\)下的纯液体物质B的状态
• 理想稀溶液:
  • 溶液温度T、标准压力\(p^\theta\)下，将亨利定律\(p_B=K_{x,B}x_B\)外延至\(x_B=1\)的溶质B的假想状态
  • 溶液温度T、标准压力\(p^\theta\)下，将亨利定律\(p_B=K_{m,B}^{'}m_B\)外延至\(m_B=1\ mol\cdot{kg^{-1}}\)的溶质B的假想状态
  • 溶液温度T、标准压力\(p^\theta\)下，将亨利定律\(p_B=K_{c,B}^{'}c_B\)外延至\(c_B=1\ mol\cdot{L^{-1}}\)的溶质B的假想状态