# Josh Testing 1

###### This wiki is incomplete.

PS: Here are examples of great wiki pages — Fractions, Chain Rule, and Vieta Root Jumping

\(\SI{1.5}{\electronvolt}\)

\(\SI{4}{\electronvolt}\)

**Ideal gas law**

The ideal gas law states that the pressure, volume, temperature, and number of molecules in a gas are related by \[\boxed{PV=Nk_BT},\] where \(k_B\) is Boltzmann's constant.

It holds for monatomic gases but requires more serious modification for gases that are diatomic, triatomic, etc. It can be derived from many starting places including the impulse momentum theorem, the partition function, and the absence of correlated motions. It accounts for a number of individual experimental observations that were made prior:

- for constant \(T\) and \(N,\) \(P\) and \(V\) are inversely proportional.
- for constant \(P\) and \(N,\) \(V\) and \(T\) are directly proportional.
- for constant \(V\) and \(N,\) \(P\) and \(T\) are directly proportional.

**Taylor expansion**

The Taylor series representation of a function is given by

\[f(x) = f(x_0) + f^\prime(x)\rvert_{x = x_0}\left(x - x_0\right) + \frac{1}{2!} f^{\prime\prime}(x)\rvert_{x = x_0}\left(x - x_0\right)^2 + \ldots.\]

which can be found through integration by parts.

Unwieldy analytic expressions can be approximated by expanding into a Taylor series of a small parameter (much less than 1) and keeping the first few terms. For example \(1/\sqrt{1-x} \approx 1 + x/2,\) when \(x \ll 1.\)

The proper time \(\tau\) is given by \(\tau^2 = \left(c\Delta t\right)^2 - \Delta x^2 - \Delta y^2 - \Delta z^2\) and is ensured by the Lorentz transformation to be frame independent. It is equal to the amount of time that passes between two events in their own rest frame, e.g. onboard a spaceship.

\(\tau^2\) classifies spacetime into events that are in the potential future and past or the absolute elsewhere relative to another event. In particular, \(\tau^2\) is zero for trajectories of lightbeams. When \(\tau^2 > 0\) the interval is timelike separated and then \(\tau^2<0\) it is spacelike separate. All causal sequences of events are timelike separated.

The work-energy theorem states that the energy needed to move a particle along a path \(\gamma\) under the force \(\mathbf{F}\) is given by \(W_\gamma = \int_\gamma \mathbf{F}\cdot d\mathbf{x}.\)) Simple manipulations of this result yield expressions for kinetic energy (by making the replacement \(F\rightarrow m\dot{v}\) and evaluating the integral), power (by re-expressing \(dx = dx/dv\cdot dv\) using the chain rule), etc. It is also used in numerical simulations to estimate the free-energy difference between states, e.g. in drug design.

Show me the \(\color{blue}{\textrm{money}}\)

Rebuttal: If \(m\) and \(n\) are factors of \( a-b,\) then they both show up in the factorization of \( a-b,\) so their product shows up too. So \(mn|(a-b).\)

Reply: This is only true if \(m\) and \(n\) are relatively prime. In the above example, \[ 13-1 = 12 = {\bf 4} \cdot 3 = {\bf 6} \cdot 2, \] but \( 4\) and \( 6\) don't show up in the same factorization of \( a-b.\)

Rebuttal: The result is true in lots of cases. For instance, if \( m\) and \(n\) are distinct primes, the result is true.

Reply: This is correct, but the statement is still false, since it does not hold for all \(m,n.\)

#### Contents

## Section One

A

cationis a positively charged atom or molecule.

If we strip a neutral \(\ce{Na}\) atom of one electron, is it an anion or a cation?

As the \(\ce{Na}\) atom starts off neutral, it has an equal number of protons and electrons. Stripping off one electron leads to an imbalance of one more proton relative to electrons, so that is it positively charged.

## Section 2: Respiration

Respiration procedes according to the following reaction

\[\ce{C6H12O6 + 6O2 -> 6CO2 + 6H2O + energy}.\]