Equation Sheets
Equation Sheets

CHEM 145 Honors General Chemistry I

  
Contents
Quantity Unit Definition
Pressure N/m2\mathrm{N/m^{2}} P=FAP = \dfrac{F}{A}
Mole fraction of aa - Xa=nantotalX_{a} = \dfrac{n_{a}}{n_{\mathrm{total}}}
Partial pressure of aa N/m2\mathrm{N/m^{2}} Pa=XaPtotalP_{a} = X_{a}P_{\mathrm{total}}
Pressure-volume work J\mathrm{J} W=−PΔVW = -P\Delta V
Compressibility of gas - z=PVnRTz = \dfrac{PV}{nRT}
Description Equations
Ideal gas law PV=nRTPV = nRT
Ideal gas constant and Boltzmann constant R=NAkBR = N_{A}k_{B}
Average kinetic energy of one gas molecule ε=32kBT\varepsilon = \dfrac{3}{2}k_{B}T
Root mean square speed gas vRMS=3RTM=3kBTmv_{\mathrm{RMS}} = \sqrt{\dfrac{3RT}{\mathcal{M}}} = \sqrt{\dfrac{3k_{B}T}{m}}
Mean speed of gas vˉ=8RTπM=8kBTπm\bar{v} = \sqrt{\dfrac{8RT}{\pi\mathcal{M}}} = \sqrt{\dfrac{8k_{B}T}{\pi m}}
Most probable speed of gas vmp=2RTM=2kBTmv_{\mathrm{mp}} = \sqrt{\dfrac{2RT}{\mathcal{M}}} = \sqrt{\dfrac{2k_{B}T}{m}}
Maxwell-Boltzmann speed distribution f(v)=4π(m2πkBT)v2exp⁡(−mv22kBT)f(v) = 4\pi \left(\dfrac{m}{2\pi k_{B}T}\right) v^{2} \exp\left(\dfrac{-mv^{2}}{2k_{B}T}\right)
Van der Waals equation of state (P+an2V2)(V−bn)=nRT(P + a\dfrac{n^{2}}{V^{2}})(V - bn) = nRT
Lennard-Jones potential VLJ=4ε((σR)12−(σR)6)V_{\mathrm{LJ}} = 4\varepsilon \left(\left(\dfrac{\sigma}{R}\right)^{12} - \left(\dfrac{\sigma}{R}\right)^{6} \right)
Description Equations
Molecule collision rate with wall Zw∝14NVvˉA=14NV8RTπMAZ_{w} \propto \dfrac{1}{4}\dfrac{N}{V}\bar{v}A = \dfrac{1}{4}\dfrac{N}{V}\sqrt{\dfrac{8RT}{\pi\mathcal{M}}}A
Graham’s law of effusion rate of effusion of Arate of effusion of B=NANBMBMA\dfrac{\text{rate of effusion of A}}{\text{rate of effusion of B}} = \dfrac{N_{\mathrm{A}}}{N_{\mathrm{B}}} \sqrt{\dfrac{\mathcal{M}_{\mathrm{B}}}{\mathcal{M}_{\mathrm{A}}}}
Molecule-molecule collision rate Z1=4NVd2πRTMZ_{1} = 4\dfrac{N}{V}d^{2}\sqrt{\dfrac{\pi RT}{\mathcal{M}}}
Mean free path λ=vˉZ1=V2πd2N\lambda = \dfrac{\bar{v}}{Z_{1}} = \dfrac{V}{\sqrt{2}\pi d^{2}N}
Mean square displacement of diffusion in 3D Δr‾2=6Dt\overline{\Delta r}^{2} = 6Dt
Gas diffusion constant D=38RTπMVNd2D = \dfrac{3}{8} \sqrt{\dfrac{RT}{\pi\mathcal{M}}} \dfrac{V}{Nd^{2}}
Quantity Unit Definition
Electrostatic force N\mathrm{N} F=q1q24πεr2F = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r^{2}}
Electrostatic potential energy N\mathrm{N} V=q1q24πεrV = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r}
Dipole moment C⋅m\mathrm{C\cdot m} μ=qd\mu = qd
Polarizability C⋅m2/V\mathrm{C\cdot m^{2}/V} α=μE\alpha = \dfrac{\mu}{E}
Induced dipole moment C⋅m\mathrm{C\cdot m} μ∗=αE\mu^{*} = \alpha E
Description Equations
Ion-ion interactions E∝q1q2rE \propto \dfrac{q_{1}q_{2}}{r}
Ion-dipole interactions E∝−qμr2E \propto -\dfrac{q\mu}{r^{2}}
Dipole-dipole interactions E∝−μ1μ2r3E \propto -\dfrac{\mu_{1}\mu_{2}}{r^{3}}
Induced-dipole-induced-dipole (London) E∝−α1α2r6E \propto -\dfrac{\alpha_{1}\alpha_{2}}{r^{6}}
Dipole-induced-dipole E∝−μ12α2r6E \propto -\dfrac{\mu_{1}^{2}\alpha_{2}}{r^{6}}
Rotating fixed dipole (Keesom) E∝−μ12μ22r6E \propto -\dfrac{\mu_{1}^{2}\mu_{2}^{2}}{r^{6}}
Quantity Unit Definition
Specific heat capacities J⋅kg−1⋅K−1\mathrm{J \cdot kg^{-1} \cdot K^{-1}} q=mcsΔT=ncˉΔTq = mc_{s}\Delta T = n\bar{c}\Delta T
Heat capacity J/K\mathrm{J/K} q=CΔTq = C\Delta T
Molar heat capacities J⋅mol−1⋅K−1\mathrm{J \cdot mol^{-1} \cdot K^{-1}} qV=ncVΔTq_{V} = nc_{V}\Delta T
qP=ncPΔTq_{P} = nc_{P}\Delta T
Enthalpy J\mathrm{J} H=U+PVH = U + PV
Description Equations
First law of thermodynamics ΔU=q+w\Delta U = q + w
dU=dq+dwdU = \cancel{d}q + \cancel{d}w
ΔUuniv=ΔUsys+ΔUsurr=0\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} + \Delta U_{\mathrm{surr}} = 0
Enthalpy change qP=Δ(U+PV)=ΔHq_{P} = \Delta (U + PV) = \Delta H
Molar heat capacity of monoatomic ideal gas at constant volume cV=32Rc_{V} = \dfrac{3}{2}R
Molar heat capacity of any ideal gas at constant pressure cP=cV+R=52Rc_{P} = c_{V} + R = \dfrac{5}{2}R
Internal energy change of any ideal gas ΔU=ncVΔT\Delta U = nc_{V}\Delta T
Enthalpy change of any ideal gas ΔH=ncPΔT=ΔU+Δ(PV)=ncVΔT+nRΔT\begin{aligned}\Delta H &= nc_{P}\Delta T \cr &= \Delta U + \Delta(PV) \cr &= nc_{V}\Delta T + nR\Delta T\end{aligned}
Hess’s law ΔH∘=∑iprodniΔHi∘−∑jreactnjΔHj∘\Delta H^{\circ} = \sum\limits_{i}^{\text{prod}} n_{i}\Delta H_{i}^{\circ} - \sum\limits_{j}^{\text{react}} n_{j}\Delta H_{j}^{\circ}
Molality b=nsolutemsolventb = \dfrac{n_{\mathrm{solute}}}{m_{\mathrm{solvent}}}
Boiling point elevation
(ii is vant’s Hoff dissociation factor)		 ΔTboil=ibKboil\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}
Freezing point depression
(ii is vant’s Hoff dissociation factor)		 ΔTfreeze=ibKfreeze\Delta T_{\mathrm{freeze}} = ibK_{\mathrm{freeze}}
Reversible isothermal process of ideal gas ΔT=0\Delta T = 0
ΔU=0\Delta U = 0
ΔH=0\Delta H = 0
w=−∫V1V2P dV=−nRTln⁡V2V1w = -\displaystyle\int_{V_{1}}^{V_{2}} P \ dV = -nRT \ln\dfrac{V_{2}}{V_{1}}
q=−wq = -w
Reversible adiabatic process of ideal gas q=0q = 0
ΔU=ncVΔT=w\Delta U = nc_{V}\Delta T = w
ΔH=ncPΔT\Delta H = nc_{P}\Delta T
γ=cPcV\gamma = \dfrac{c_{P}}{c_{V}}
T1V1γ−1=T2V2γ−1T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}
P1V1γ=P2V2γP_{1}V_{1}^{\gamma} = P_{2}V_{2}^{\gamma}
Description Equations
Entropy S=kBln⁡ΩS = k_{B} \ln\Omega
Entropy change ΔS=∫ifdqrevT\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T}
ΔSsys\Delta S_{\mathrm{sys}} for reversible isothermal process ΔS=∫ifdqrevT=1T∫ifdqrev=qrevT\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T} = \dfrac{1}{T} \displaystyle\int_{i}^{f} dq_{\mathrm{rev}} = \dfrac{q_{\mathrm{rev}}}{T}
ΔSsys\Delta S_{\mathrm{sys}} for reversible isothermal process - compression/expansion of ideal gas qrev=nRTln⁡(V2V1)q_{\mathrm{rev}} = nRT \ln \left( \dfrac{V_{2}}{V_{1}} \right)
ΔS=nRln⁡(V2V1)\Delta S = nR \ln \left( \dfrac{V_{2}}{V_{1}} \right)
ΔSsys\Delta S_{\mathrm{sys}} for reversible isothermal process - phase transitions qrev=ΔHfusq_{\mathrm{rev}} = \Delta H_{\mathrm{fus}}
ΔSfus=qrevTfus=ΔHfusTfus\Delta S_{\mathrm{fus}} = \dfrac{q_{\mathrm{rev}}}{T_{\mathrm{fus}}} = \dfrac{\Delta H_{\mathrm{fus}}}{T_{\mathrm{fus}}}
ΔSsys\Delta S_{\mathrm{sys}} for reversible adiabatic process q=0q = 0
ΔS=0\Delta S = 0
ΔSsys\Delta S_{\mathrm{sys}} for reversible isochoric process ΔV=0\Delta V = 0
dqrev=ncVdTdq_{\mathrm{rev}} = nc_{V}dT
ΔS=ncV∫T1T2dTT=ncVln⁡(T2T1)\Delta S = nc_{V} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{V} \ln \left( \dfrac{T_{2}}{T_{1}} \right)
ΔSsys\Delta S_{\mathrm{sys}} for reversible isobaric process ΔP=0\Delta P = 0
dqrev=ncPdTdq_{\mathrm{rev}} = nc_{P}dT
ΔS=ncP∫T1T2dTT=ncPln⁡(T2T1)\Delta S = nc_{P} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{P} \ln \left( \dfrac{T_{2}}{T_{1}} \right)
Entropy change of surrounding ΔSsurr=−ΔHsysTsurr\Delta S_{\mathrm{surr}} = \dfrac{-\Delta H_{\mathrm{sys}}}{T_{\mathrm{surr}}}
Second law of thermodynamics ΔS≥qrevT\Delta S \geq \dfrac{q_{\mathrm{rev}}}{T}
Enthalpy of spontaneous process ΔStotal=ΔSsys+ΔSsurr>0\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} > 0
Standard molar entropy S∘=∫0K298.15KcPTdT+ΔS(phase changes between 0K and 298.15K)S^{\circ} = \displaystyle\int_{0K}^{298.15\mathrm{K}} \dfrac{c_{P}}{T} dT + \Delta S \text{(phase changes between 0K and 298.15K)}
Gibbs free energy for reaction at constant temperature ΔG=ΔH−TΔS\Delta G = \Delta H - T\Delta S
Efficiency of Carnot engines ε=work on surroundingheat into system=Thigh−TlowThigh=1−TlowThigh\begin{aligned}\varepsilon &= \dfrac{\text{work on surrounding}}{\text{heat into system}} \cr &= \dfrac{T_{\mathrm{high}} - T_{\mathrm{low}}}{T_{\mathrm{high}}} = 1-\dfrac{T_{\mathrm{low}}}{T_{\mathrm{high}}}\end{aligned}
Relationship between heat and temperature in Carnot cycle qhighThigh+qlowTlow=0\dfrac{q_{\mathrm{high}}}{T_{\mathrm{high}}} + \dfrac{q_{\mathrm{low}}}{T_{\mathrm{low}}} = 0
Work done by Carnot cycle in one cycle wcycle=−nR(Thot−Tcold)ln⁡VBVAw_{\mathrm{cycle}} = -nR(T_{\mathrm{hot}} - T_{\mathrm{cold}}) \ln\dfrac{V_{B}}{V_{A}}
Description Equations
Law of mass action - partial pressure K=∏j(Pproduct j/Pref)eqbj∏i(Preactant i/Pref)eqaiK = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})_{eq}^{a_{i}}}
Law of mass action - concentration K=∏j(cproduct j/cref)eqbj∏i(creactant i/cref)eqaiK = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})_{eq}^{a_{i}}}
Gibbs free energy of isothermal reactions ΔG=−TΔS=nRTln⁡P2P1\Delta G = -T\Delta S = nRT\ln\dfrac{P_{2}}{P_{1}}
Equilibrium expression: relationship between Gibbs free energy and equilibrium constant (gas phase reaction) ΔG∘=−RTln⁡K\Delta G^{\circ} = -RT\ln K
Equilibrium expression: alternative form ln⁡K=−ΔG∘RT=−ΔH∘RT+ΔSR\ln K = -\dfrac{\Delta G^{\circ}}{RT} = - \dfrac{\Delta H^{\circ}}{RT} + \dfrac{\Delta S}{R}
Change in Gibbs free energy at non-standard conditions ΔG=ΔG∘+RTln⁡Q=RTln⁡QK\Delta G = \Delta G^{\circ} + RT\ln Q = RT\ln\dfrac{Q}{K}
Reaction quotient - partial pressure Q=∏j(Pproduct j/Pref)bj∏i(Preactant i/Pref)aiQ = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})^{a_{i}}}
Reaction quotient - concentration Q=∏j(cproduct j/cref)bj∏i(creactant i/cref)aiQ = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})^{a_{i}}}
vant’s Hoff equation
temperature dependence of equilibrium constant of a reaction		 ln⁡K2K1=−ΔH∘R(1T2−1T1)\ln\dfrac{K_{2}}{K_{1}} = -\dfrac{\Delta H^{\circ}}{R} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)
Clapeyron equation
for two phase in equilibrium, construct phase diagram by finding change of pressure as a function of temperature		 (dPdT)eq=ΔSΔV=ΔHTΔV\left( \dfrac{dP}{dT} \right)_{eq} = \dfrac{\Delta S}{\Delta V} = \dfrac{\Delta H}{T\Delta V}
Clausius-Clapeyron equation
temperature dependence of vapor pressure for condensed phase and gas phase in equilibrium		 ln⁡P2P1=−ΔHvapnR(1T2−1T1)\ln\dfrac{P_{2}}{P_{1}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nR} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)
Clausius-Clapeyron equation
for P1=P∘;ΔS∘=ΔH∘T∘P_{1} = P^{\circ}; \Delta S^{\circ} = \dfrac{\Delta H^{\circ}}{T^{\circ}}		 ln⁡P2P∘=−ΔHvapnRT2+ΔSvap∘nR\ln\dfrac{P_{2}}{P^{\circ}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nRT_{2}} + \dfrac{\Delta S^{\circ}_{\mathrm{vap}}}{nR}
Updated on 2021-08-17
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