Hamilton's Equation
Hamilton's Equation - Related Videos
03:10
Chemical Equations
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2020
Chemical equations represent the identities and relative quantities of substances involved in a chemical reaction. The substances undergoing reaction are called reactants, and their formulas are placed on the left side of the equation. The substances generated by the reaction are called products, and their formulas are placed on the right side of the equation. Plus signs (+) separate individual reactant and product formulas, and an arrow (→) separates the reactant and product (left and right)...
02:55
Thermochemical Equations
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2020
For a chemical reaction (the system) carried out at constant pressure – with the only work done caused by expansion or contraction – the enthalpy of reaction (also called the heat of reaction, ΔHrxn) is equal to the heat exchanged with the surroundings (qp). The change in enthalpy is an extensive property, and it depends on the amounts of the reactants participating in the reaction (or the number of moles of reactants). The change in enthalpy is specific to the reaction, and the physical...
02:59
The Nernst Equation
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2020
Nonstandard Reaction Conditions The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eºcell = +0.291 V and ΔG° = −56.2 kJ is spontaneous. Discharge of this cell, however, results in a change in the reactant concentration and a steady decrease of the cell potential. At...
02:48
Henderson-Hasselbalch Equation
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2020
The ionization-constant expression for a solution of a weak acid can be written as: Rearranging to solve for [H3O+] yields: Taking the negative logarithm of both sides of this equation gives which can be written as where pKa is the negative of the logarithm of the ionization constant of the weak acid (pKa = −log Ka). This equation relates the pH, the ionization constant of a weak acid, and the concentrations of the weak conjugate acid-base pair in a buffered solution. Scientists often use...
02:35
Clausius-Clapeyron Equation
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2020
The equilibrium between a liquid and its vapor depends on the temperature of the system; a rise in temperature causes a corresponding rise in the vapor pressure of its liquid. The Clausius-Clapeyron equation gives the quantitative relation between a substance’s vapor pressure (P) and its temperature (T); it predicts the rate at which vapor pressure increases per unit increase in temperature. where ΔHvap is the enthalpy of vaporization for the liquid, R is the gas constant, and A is a constant...
FAQs
The three variables combine multiplicatively and comparatively: relatedness scales the recipient’s benefit, producing rb, and that value is weighed against the actor’s cost, c. A larger benefit or greater relatedness strengthens the evolutionary case for helping, whereas a larger cost weakens it. The equation therefore evaluates the balance between effects on the actor and recipient.
A cost to the actor does not determine the outcome by itself because the recipient may gain a benefit that matters through shared genes. When the relatedness-weighted benefit, rb, exceeds the actor’s cost, c, selection can favor the behavior. This logic explains why cooperation or helping may evolve even when it reduces the actor’s immediate advantage.
Hamilton’s equation provides the condition linking these two ideas. Kin selection focuses on how helping relatives can affect evolutionary change, while inclusive fitness accounts for the actor’s own reproductive contribution together with effects on relatives. The relationship term, r, connects the recipient’s benefit to shared genes, showing why assistance to relatives can contribute to gene transmission.
Researchers can identify the actor’s cost, estimate the recipient’s benefit, and assess the genetic relatedness between them. They then compare rb with c to determine whether the observed pattern meets the condition favoring the behavior. This framework gives studies of cooperation and helping a common way to organize the relevant evolutionary variables.
The framework applies to questions about cooperation, altruism, parental care, and social behavior. It is especially useful when one individual’s action affects another individual and researchers need to evaluate both the cost to the actor and the recipient’s benefit. Studies of social insects also use this perspective to examine how helping behavior may evolve.
The inequality identifies a condition under which natural selection favors a trait that benefits another individual despite a cost to its actor. It does not treat benefit or cost alone as decisive; the recipient’s benefit must first be weighted by relatedness. Researchers can therefore use the comparison to interpret whether a social behavior fits the kin-selection framework.