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GATE 2027 XE5 & CIPET Master Revision Formula Sheet

High-density subject-wise formula cheat sheet for rapid exam revision & shop-floor reference (IIT Madras Aligned).

Showing 32 Formulas (28 GATE XE5 Tagged)PolymerHub Official Curriculum Sheet
IDFormula NameSubjectTypeCanonical Math EquationVariable Definitions & SI Units
PH-FORM-CHEM-001Carothers Equation (Step-Growth Polymerization)Polymer ChemistryT3
Xn=22−p⋅favgX_n = \frac{2}{2 - p \cdot f_{avg}}
X_n (dimensionless), p (dimensionless), f_{avg} (dimensionless)
PH-FORM-CHEM-002Mayo-Lewis Copolymerization EquationPolymer ChemistryT3
d[M1]d[M2]=[M1][M2]⋅r1[M1]+[M2]r2[M2]+[M1]\frac{d[M_1]}{d[M_2]} = \frac{[M_1]}{[M_2]} \cdot \frac{r_1 [M_1] + [M_2]}{r_2 [M_2] + [M_1]}
d[M_1]/d[M_2] (molar ratio), [M_1]/[M_2] (molar ratio), r_1 (dimensionless), r_2 (dimensionless)
PH-FORM-CHEM-003Fox Equation (Copolymer & Blend Tg)Polymer ChemistryT2
1Tg=w1Tg1+w2Tg2\frac{1}{T_g} = \frac{w_1}{T_{g1}} + \frac{w_2}{T_{g2}}
T_g (K (°C)), w_1 (fraction), T_{g1} (K), w_2 (fraction), T_{g2} (K)
PH-FORM-PROC-001Injection Mould Clamping Tonnage FormulaPolymer ProcessingT5
Fclamp=Pcavity⋅Aprojected1000⋅SFF_{clamp} = \frac{P_{cavity} \cdot A_{projected}}{1000} \cdot SF
F_{clamp} (Metric Tonnes), P_{cavity} (bar (kg/cm²)), A_{projected} (cm²), SF (dimensionless)
PH-FORM-PROC-0021D Fourier Transient Cooling Time FormulaPolymer ProcessingT5
tc=h2π2⋅αln⁡[8π2⋅Tmelt−TmoldTeject−Tmold]t_c = \frac{h^2}{\pi^2 \cdot \alpha} \ln\left[ \frac{8}{\pi^2} \cdot \frac{T_{melt} - T_{mold}}{T_{eject} - T_{mold}} \right]
t_c (seconds), h (mm), \alpha (mm²/s), T_{melt} (°C), T_{mold} (°C), T_{eject} (°C)
PH-FORM-RHEO-001Ostwald-de Waele Power-Law Shear Thinning ModelPolymer RheologyT4
τ=K⋅γ˙n⇒η(γ˙)=K⋅γ˙n−1\tau = K \cdot \dot{\gamma}^n \quad \Rightarrow \quad \eta(\dot{\gamma}) = K \cdot \dot{\gamma}^{n-1}
\tau (Pa (N/m²)), \eta (Pa·s), K (Pa·s^n), \dot{\gamma} (s⁻¹), n (dimensionless)
PH-FORM-TEST-001Engineering Tensile Stress, Strain & Young’s ModulusPolymer Testing & CharacterizationT6
σ=FA0,ϵ=ΔLL0,E=σϵ\sigma = \frac{F}{A_0}, \quad \epsilon = \frac{\Delta L}{L_0}, \quad E = \frac{\sigma}{\epsilon}
\sigma (MPa (N/mm²)), \epsilon (mm/mm (%)), E (MPa (GPa)), F (N), A_0 (mm²), \Delta L (mm), L_0 (mm)
PH-FORM-TOOL-001Hagen-Poiseuille Circular Runner Pressure DropMould & Die DesignT5
ΔPrunner=128⋅μ⋅Q⋅Lπ⋅d4\Delta P_{runner} = \frac{128 \cdot \mu \cdot Q \cdot L}{\pi \cdot d^4}
\Delta P (bar (MPa)), \mu (Pa·s), Q (cm³/s), L (mm), d (mm)
PH-FORM-RUBB-001Flory-Rehner Vulcanized Rubber Crosslink DensityRubber & Elastomer TechnologyT2
νe=−ln⁡(1−v2)+v2+χ⋅v22V1⋅(v21/3−v22)\nu_e = - \frac{\ln(1 - v_2) + v_2 + \chi \cdot v_2^2}{V_1 \cdot \left( v_2^{1/3} - \frac{v_2}{2} \right)}
\nu_e (mol/cm³), v_2 (fraction), V_1 (cm³/mol), \chi (dimensionless)
PH-FORM-COMP-001Voigt Longitudinal Rule of MixturesPolymer Composites & Fiber EngineeringT8
E11=Vf⋅Ef+(1−Vf)⋅EmE_{11} = V_f \cdot E_f + (1 - V_f) \cdot E_m
E_{11} (GPa), V_f (fraction), E_f (GPa), E_m (GPa)
PH-FORM-PACK-001Fickian Film Oxygen Transmission Rate (OTR)Plastic Packaging EngineeringT7
OTR=PO2⋅ΔptOTR = \frac{P_{O2} \cdot \Delta p}{t}
OTR (cc/(m²·day)), P_{O2} (cc·mil/(m²·day·atm)), \Delta p (atm), t (mil (1 mil = 25.4 μm))
PH-FORM-SUST-001MoEFCC EPR Plastic Credit Obligation FormulaSustainable Plastics & Circular EconomyT11
EPRobligation=Msales⋅ftarget⋅(1−wrecycled)EPR_{obligation} = M_{sales} \cdot f_{target} \cdot (1 - w_{recycled})
EPR_{obligation} (Metric Tonnes), M_{sales} (Metric Tonnes), \%_{target} (fraction), w_{recycled} (fraction)
PH-FORM-COLOR-001CIE L*a*b* Delta-E 1976 Color Difference FormulaColor Science & Masterbatch TechnologyT9
ΔEab∗=(ΔL∗)2+(Δa∗)2+(Δb∗)2\Delta E^*_{ab} = \sqrt{ (\Delta L^*)^2 + (\Delta a^*)^2 + (\Delta b^*)^2 }
\Delta E^* (units), \Delta L^* (units), \Delta a^* (units), \Delta b^* (units)
PH-FORM-ENTR-001Plastics Factory Break-Even Point (BEP) AnalysisEntrepreneurship & Factory SetupT10
BEPunits=FfixedPselling−Vvariable⇒BEPsales=Ffixed1−VPBEP_{units} = \frac{F_{fixed}}{P_{selling} - V_{variable}} \quad \Rightarrow \quad BEP_{sales} = \frac{F_{fixed}}{1 - \frac{V}{P}}
BEP_{units} (kg/year), F_{fixed} (₹ (Rupees)), P_{selling} (₹/kg), V_{variable} (₹/kg)
PH-FORM-CHEM-004Flory-Fox Equation (Tg Dependence on Molecular Weight)Polymer ChemistryT2
Tg=Tg,∞−KffM‾nT_g = T_{g,\infty} - \frac{K_{ff}}{\overline{M}_n}
T_g (°C (or K)), T_{g,infty} (°C), K_{ff} (K·g/mol), \overline{M}_n (g/mol)
PH-FORM-CHEM-005Mark-Houwink-Sakurada Equation (Intrinsic Viscosity to MW)Polymer ChemistryT9
[η]=K⋅M‾va[\eta] = K \cdot \overline{M}_v^a
[\eta] (dL/g), K (dL/g), a (dimensionless), \overline{M}_v (g/mol)
PH-FORM-PROC-003Single-Screw Extruder Drag Flow Rate (Throughput Capacity)Polymer ProcessingT5
Qd=12π2D2HNsin⁡θcos⁡θQ_d = \frac{1}{2} \pi^2 D^2 H N \sin\theta \cos\theta
Q_d (cm³/s (or kg/h)), D (mm), H (mm), N (RPM), \theta (degrees)
PH-FORM-MOULD-002Injection Mould Cavity Shrinkage Dimension Sizing FormulaMould & Die DesignT5
Dc=Dpcdot(1+Sl)D_c = D_p cdot (1 + S_l)
D_c (mm), D_p (mm), S_l (fraction (mm/mm))
PH-FORM-RHEO-002Williams-Landel-Ferry (WLF) Shift Factor EquationPolymer Rheology & Melt FlowT4
log⁡aT=−C1(T−Tg)C2+(T−Tg)\log a_T = \frac{-C_1 (T - T_g)}{C_2 + (T - T_g)}
\log a_T (dimensionless), T (°C (or K)), T_g (°C (or K)), C_1 (dimensionless), C_2 (K)
PH-FORM-TEST-002DSC Degree of Crystallinity Percentage FormulaPolymer Testing & Quality ControlT9
Xc=ΔHm−ΔHccΔHm0⋅(1−wf)×100X_c = \frac{\Delta H_m - \Delta H_{cc}}{\Delta H_m^0 \cdot (1 - w_f)} \times 100
X_c (%), \Delta H_m (J/g), \Delta H_{cc} (J/g), \Delta H_m^0 (J/g), w_f (fraction)
PH-FORM-PACK-002Water Vapor Transmission Rate (WVTR) Through Barrier FilmsPlastic Packaging EngineeringT7
WVTR=PH2O⋅ΔpH2OlWVTR = \frac{P_{H_2O} \cdot \Delta p_{H_2O}}{l}
WVTR (g/(m²·day)), P_{H_2O} (g·mil/(100 in²·day) or g·mm/(m²·day)), l (μm (microns))
PH-FORM-SUST-002Biobased Carbon Content Percentage Formula (ASTM D6866)Sustainable Plastics & Circular EconomyT11
Xbio=14Csample14Cmodern×100  [%]X_{bio} = \frac{^{14}\text{C}_{sample}}{^{14}\text{C}_{modern}} \times 100 \; [\%]
\text{Biobased}\% (% of Total Organic Carbon), ^{14}\text{C}_{sample} (pMC (percent Modern Carbon)), ^{14}\text{C}_{modern} (pMC)
PH-FORM-RECY-001Mechanical Recycling Mass Balance & Washing Flake YieldRecycling Technology & ProcessingT11
Yflake=Mclean_flakeMraw_bales×100Y_{flake} = \frac{M_{clean\_flake}}{M_{raw\_bales}} \times 100
Y_{flake} (%), M_{clean\_flake} (Tonnes), M_{raw\_bales} (Tonnes)
PH-FORM-RUBB-002Mooney-Rivlin Strain Energy Density Equation (Hyperelasticity)Rubber & Elastomer TechnologyT6
W=C10(I1−3)+C01(I2−3)W = C_{10} (I_1 - 3) + C_{01} (I_2 - 3)
W (MJ/m³ (or MPa)), C_{10} (MPa), C_{01} (MPa), I_1 (dimensionless), I_2 (dimensionless)
PH-FORM-COMP-002Reuss Transverse Rule of Mixtures Modulus (Inverse / Series Model)Polymer Composites & Fiber EngineeringT8
1E2=VfEf+1−VfEm\frac{1}{E_2} = \frac{V_f}{E_f} + \frac{1 - V_f}{E_m}
E_2 (GPa), V_f (fraction), E_f (GPa), E_m (GPa)
PH-FORM-ADD-001Parts Per Hundred Resin (PHR) to Weight Percentage ConversionAdditives & Compounding TechnologyT1
wi=PHRi100+∑jPHRj×100w_i = \frac{\text{PHR}_i}{100 + \sum_{j} \text{PHR}_j} \times 100
w_i (wt%), \text{PHR}_i (PHR), \sum \text{PHR}_j (PHR)
PH-FORM-LCA-001Global Warming Potential (GWP) Carbon Footprint per kg ResinLife Cycle Assessment (LCA)T11
GWP=∑i(mi⋅EFi)+(Eelec⋅EFgrid)\text{GWP} = \sum_{i} (m_i \cdot EF_i) + (E_{elec} \cdot EF_{grid})
\text{GWP} (kg CO2e / kg polymer), m_i \cdot EF_i (kg CO2e / kg), E_{elec} (kWh / kg), EF_{grid} (kg CO2e / kWh)
PH-FORM-MED-001Higuchi Model for Fickian Drug Release from Polymeric MatricesMedical Plastics & BiocompatibilityT7
Mt=AD⋅(2C0−Cs)⋅Cs⋅tM_t = A \sqrt{D \cdot (2C_0 - C_s) \cdot C_s \cdot t}
M_t / A (mg/cm²), C_0 (mg/cm³), C_s (mg/cm³), D (cm²/s), t (hours)
PH-FORM-DIGI-001Cavity Pressure Integral (Mould Cavity Energy Work Done)Digital Twins, Industry 4.0 & AIT12
Wcav=∫tinjtfreezePcav(t) dtW_{cav} = \int_{t_{inj}}^{t_{freeze}} P_{cav}(t) \, dt
W_{cav} (bar·s), P_{peak} (bar), t_{pack} (seconds)
PH-FORM-BIO-001Monod Specific Microbial Growth Rate for PHA BiopolymersBioprocessing & Microbial FermentationT3
μ=μmax[S]Ks+[S]\mu = \mu_{max} \frac{[S]}{K_s + [S]}
\mu (h⁻¹), \mu_{max} (h⁻¹), [S] (g/L), K_s (g/L)
PH-FORM-ROBO-0013-Axis Cartesian Robot Demoulding Take-Out Time FormulaRobotics & Automation in PlasticsT12
ttakeout=tenter+tgrip+tstrip+texitt_{takeout} = t_{enter} + t_{grip} + t_{strip} + t_{exit}
t_{takeout} (seconds), t_{enter} (seconds), t_{grip} (seconds), t_{strip} (seconds), t_{exit} (seconds)
PH-FORM-NANO-001Electrical Percolation Threshold in Carbon Nanotube NanocompositesPolymer Nanotechnology & NanocompositesT6
σ=σ0(ϕ−ϕc)tfor ϕ>ϕc\sigma = \sigma_0 (\phi - \phi_c)^t \quad \text{for } \phi > \phi_c
\sigma (S/m), \sigma_0 (S/m), \phi (volume fraction), \phi_c (volume fraction), t (dimensionless)