diff --git a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md
index 894bc36..524dff9 100755
--- a/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md
+++ b/uni/mmme/1048_thermodynamics_and_fluid_mechanics/thermodynamics.md
@@ -116,211 +116,6 @@ c_p &= \frac{c_p}{\gamma} + R \\
c_p &= \frac{\gamma}{\gamma -1} R
\end{align*}
-
-
-
-
-## Properties of State
-
-*State* is defined as the condition of a system as described by its properties.
-The state may be identified by certain observable macroscopic properties.
-These properties are the *properties of state* and they always have the same values for a given
-state.
-
-A *property* can be defined as any quantity that depends on the *state* of the system and is
-independant of the path by which the system arrived at the given state.
-Properties determining the state of a thermodynamic system are referred to as *thermodynamic
-properties* of the *state* of the system.
-
-Common properties of state are:
-
-- Temperature
-- Pressure
-- Mass
-- Volume
-
-And these can be determined by simple measurements.
-Other properties can be calculated:
-
-- Specific volume
-- Density
-- Internal energy
-- Enthalpy
-- Entropy
-
-### Intensive vs Extensive Properties
-
-In thermodynamics we distinguish between *intensive*, *extensive*, and *specific* properties:
-
-- Intensive --- properties which do not depend on mass (e.g. temperature)
-- Extensive --- properties which do depend on the mass of substance in a system (e.g. volume)
-- Specific (extensive) --- extensive properties which are reduced to unit mass of substance
- (essentially an extensive property divided by mass) (e.g. specific volume)
-
-### Units
-
-
-
-Property | Symbol | Units | Intensive | Extensive
---------------- | ------ | --------------- | --------- | ---------
-Pressure | p | Pa | Yes |
-Temperature | T | K | Yes |
-Volume | V | m$^3$ | | Yes
-Mass | m | kg | | Yes
-Specific Volume | v | m$^3$ kg$^{-1}$ | Yes |
-Density | $\rho$ | kg m$^{-3}$ | Yes |
-Internal Energy | U | J | | Yes
-Entropy | S | J K$^{-1}$ | | Yes
-Enthalpy | H | J | | Yes
-
-
-
-### Density
-
-For an ideal gas:
-
-$$\rho = \frac{p}{RT}$$
-
-### Enthalpy and Specific Enthalpy
-
-Enthalpy does not have a general physical interpretation.
-It is used because the combination $u + pv$ appears naturally in the analysis of many
-thermodynamic problems.
-
-The heat transferred to a closed system undergoing a reversible constant pressure process is equal
-to the change in enthalpy of the system.
-
-Enthalpy is defined as:
-
-$$H = U+pV$$
-
-and Specific Enthalpy:
-
-$$h = u + pv$$
-
-### Entropy and Specific Entropy
-
-Entropy is defined as the following, given that the process s reversible:
-
-$$S_2 - S_1 = \int\! \frac{\mathrm{d}Q}{T}$$
-
-#### Change of Entropy of a Perfect Gas
-
-Consider the 1st corollary of the 1st law:
-
-$$\mathrm dq + \mathrm dw = \mathrm du$$
-
-and that the process is reversible:
-
-\begin{align*}
-\mathrm ds &= \frac{\mathrm dq} T \bigg|_{rev} \\
-\mathrm dq = \mathrm ds \times T \\
-\mathrm dw &= -p\mathrm dv \\
-\end{align*}
-
-The application of the 1st corollary leads to:
-
-$$T\mathrm ds - p\mathrm dv = \mathrm du$$
-
-Derive the change of entropy
-
-\begin{align*}
-\mathrm ds &= \frac{\mathrm du}{T} + \frac{p \mathrm dv}{T} \\
-\\
-\mathrm du &= c_v \mathrm{d}T \\
-\frac p T &= \frac R v \\
-\\
-\mathrm ds &= \frac{c_v\mathrm{d}T}{T} \frac{R\mathrm dv}{v} \\
-s_2 - s_1 &= c_v\ln\left(\frac{T_2}{T_1}\right) + R\ln\left(\frac{v_2}{v_1}\right)
-\end{align*}
-
-There are two other forms of the equation that can be derived:
-
-$$s_2 - s_1 = c_v\ln\left(\frac{p_2}{p_1}\right) + c_p\ln\left(\frac{v_2}{v_1}\right)$$
-$$s_2 - s_1 = c_p\ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{p_2}{p_1}\right)$$
-
-### Heat Capacity and Specific Heat Capacity
-
-Heat capacity is quantity of heat required to raise the temperature of a system by a unit
-temperature:
-
-$$C = \frac{\mathrm{d}Q}{\mathrm{d}T}$$
-
-Specific heat capacity is the quantity of heat required to raise the temperature of a unit mass
-substance by a unit temperature:
-
-$$c = \frac{\mathrm{d}q}{\mathrm{d}T}$$
-
-
-
-
-#### Heat Capacity in Closed Systems and Internal Energy
-
-The specific heat transfer to a closed system during a reversible constant **volume** process is
-equal to the change in specific **internal energy** of the system:
-
-$$c_v = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}u}{\mathrm{d}T}$$
-
-
-
-This is because if the change in volume, $\mathrm{d}v = 0$, then the work done, $\mathrm{d}w = 0$
-also.
-So applying the (1st Corollary of the) 1st Law to an isochoric process:
-
-$$\mathrm{d}q + \mathrm{d}w = \mathrm{d}u \rightarrow \mathrm{d}q = \mathrm{d}u$$
-
-since $\mathrm{d}w = 0$.
-
-
-
-
-
-
-#### Heat Capacity in Closed Systems and Enthalpy
-
-The specific heat transfer to a closed system during a reversible constant **pressure** process is
-equal to the change in specific **enthalpy** of the system:
-
-$$c_p = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}h}{\mathrm{d}T}$$
-
-
-
-This is because given that pressure, $p$, is constant, work, $w$, can be expressed as:
-
-$$w = -\int^2_1\! p \,\mathrm{d}v = -p(v_2 - v_1)$$
-
-Applying the (1st corollary of the) 1st law to the closed system:
-
-\begin{align*}
- q + w &= u_2 - u_1 \rightarrow q = u_2 - u_1 + p(v_2 - v_1) \\
- q &= u_2 + pv_2 - (u_1 + pv_1) \\
- &= h_2 - h_1 = \mathrm{d}h \\
- \therefore \mathrm{d}q &= \mathrm{d}h
-\end{align*}
-
-
-
-
-
-
-#### Ratio of Specific Heats
-
-$c_p > c_v$ is always true.
-
-
-
-Heating a volume of fluid, $V$, at a constant volume requires specific heat $q_v$ where
-
-$$q_v = u_2 - u_1 \therefore c_v = \frac{q_v}{\Delta T}$$
-
-Heating the same volume of fluid but under constant pressure requires a specific heat $q_p$ where
-
-$$q_p =u_2 - u_1 + p(v_2-v_1) \therefore c_p = \frac{q_p}{\Delta T}$$
-
-Since $p(v_2-v_1) > 0$, $\frac{q_p}{q_v} > 1 \therefore q_p > q_v \therefore c_p > c_v$.
-
-The ratio $\frac{c_p}{c_v} = \gamma$
-
## Thermodynamic Processes and Cycles
@@ -355,7 +150,7 @@ Constant volume process
## Heat and Work
-Heat and Work are different forms of enery transfer.
+Heat and Work are different forms of energy transfer.
They are both transient phenomena and systems never possess heat or work.
Both represent energy crossing boundaries when a system undergoes a change of state.
@@ -385,10 +180,6 @@ In thermally insulated systems and isolated systems, heat transfer cannot take p
In thermally isolated systems, work transfer cannot take place.
-# Process and State Diagrams
-
-Reversible processes are represented by solid lines, and irreversible processes by dashed lines.
-
# 1st Law of Thermodynamics
The 1st Law of Thermodynamics can be thought of as:
@@ -411,7 +202,216 @@ The 1st Law of Thermodynamics can be thought of as:
> The internal energy of a closed system remains unchanged if it
> [thermally isolated](#thermally-insulated-and-isolated-systems) from its surroundings
-# Polytropic Processes
+# Properties of State
+
+*State* is defined as the condition of a system as described by its properties.
+The state may be identified by certain observable macroscopic properties.
+These properties are the *properties of state* and they always have the same values for a given
+state.
+
+A *property* can be defined as any quantity that depends on the *state* of the system and is
+independant of the path by which the system arrived at the given state.
+Properties determining the state of a thermodynamic system are referred to as *thermodynamic
+properties* of the *state* of the system.
+
+Common properties of state are:
+
+- Temperature
+- Pressure
+- Mass
+- Volume
+
+And these can be determined by simple measurements.
+Other properties can be calculated:
+
+- Specific volume
+- Density
+- Internal energy
+- Enthalpy
+- Entropy
+
+## Intensive vs Extensive Properties
+
+In thermodynamics we distinguish between *intensive*, *extensive*, and *specific* properties:
+
+- Intensive --- properties which do not depend on mass (e.g. temperature)
+- Extensive --- properties which do depend on the mass of substance in a system (e.g. volume)
+- Specific (extensive) --- extensive properties which are reduced to unit mass of substance
+ (essentially an extensive property divided by mass) (e.g. specific volume)
+
+## Units
+
+
+
+Property | Symbol | Units | Intensive | Extensive
+--------------- | ------ | --------------- | --------- | ---------
+Pressure | p | Pa | Yes |
+Temperature | T | K | Yes |
+Volume | V | m$^3$ | | Yes
+Mass | m | kg | | Yes
+Specific Volume | v | m$^3$ kg$^{-1}$ | Yes |
+Density | $\rho$ | kg m$^{-3}$ | Yes |
+Internal Energy | U | J | | Yes
+Entropy | S | J K$^{-1}$ | | Yes
+Enthalpy | H | J | | Yes
+
+
+
+## Density
+
+For an ideal gas:
+
+$$\rho = \frac{p}{RT}$$
+
+## Enthalpy and Specific Enthalpy
+
+Enthalpy does not have a general physical interpretation.
+It is used because the combination $u + pv$ appears naturally in the analysis of many
+thermodynamic problems.
+
+The heat transferred to a closed system undergoing a reversible constant pressure process is equal
+to the change in enthalpy of the system.
+
+Enthalpy is defined as:
+
+$$H = U+pV$$
+
+and Specific Enthalpy:
+
+$$h = u + pv$$
+
+## Entropy and Specific Entropy
+
+Entropy is defined as the following, given that the process s reversible:
+
+$$S_2 - S_1 = \int\! \frac{\mathrm{d}Q}{T}$$
+
+### Change of Entropy of a Perfect Gas
+
+Consider the 1st corollary of the 1st law:
+
+$$\mathrm dq + \mathrm dw = \mathrm du$$
+
+and that the process is reversible:
+
+\begin{align*}
+\mathrm ds &= \frac{\mathrm dq} T \bigg|_{rev} \\
+\mathrm dq = \mathrm ds \times T \\
+\mathrm dw &= -p\mathrm dv \\
+\end{align*}
+
+The application of the 1st corollary leads to:
+
+$$T\mathrm ds - p\mathrm dv = \mathrm du$$
+
+Derive the change of entropy
+
+\begin{align*}
+\mathrm ds &= \frac{\mathrm du}{T} + \frac{p \mathrm dv}{T} \\
+\\
+\mathrm du &= c_v \mathrm{d}T \\
+\frac p T &= \frac R v \\
+\\
+\mathrm ds &= \frac{c_v\mathrm{d}T}{T} \frac{R\mathrm dv}{v} \\
+s_2 - s_1 &= c_v\ln\left(\frac{T_2}{T_1}\right) + R\ln\left(\frac{v_2}{v_1}\right)
+\end{align*}
+
+There are two other forms of the equation that can be derived:
+
+$$s_2 - s_1 = c_v\ln\left(\frac{p_2}{p_1}\right) + c_p\ln\left(\frac{v_2}{v_1}\right)$$
+$$s_2 - s_1 = c_p\ln\left(\frac{T_2}{T_1}\right) - R\ln\left(\frac{p_2}{p_1}\right)$$
+
+## Heat Capacity and Specific Heat Capacity
+
+Heat capacity is quantity of heat required to raise the temperature of a system by a unit
+temperature:
+
+$$C = \frac{\mathrm{d}Q}{\mathrm{d}T}$$
+
+Specific heat capacity is the quantity of heat required to raise the temperature of a unit mass
+substance by a unit temperature:
+
+$$c = \frac{\mathrm{d}q}{\mathrm{d}T}$$
+
+
+
+
+### Heat Capacity in Closed Systems and Internal Energy
+
+The specific heat transfer to a closed system during a reversible constant **volume** process is
+equal to the change in specific **internal energy** of the system:
+
+$$c_v = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}u}{\mathrm{d}T}$$
+
+
+
+This is because if the change in volume, $\mathrm{d}v = 0$, then the work done, $\mathrm{d}w = 0$
+also.
+So applying the (1st Corollary of the) 1st Law to an isochoric process:
+
+$$\mathrm{d}q + \mathrm{d}w = \mathrm{d}u \rightarrow \mathrm{d}q = \mathrm{d}u$$
+
+since $\mathrm{d}w = 0$.
+
+
+
+
+
+
+### Heat Capacity in Closed Systems and Enthalpy
+
+The specific heat transfer to a closed system during a reversible constant **pressure** process is
+equal to the change in specific **enthalpy** of the system:
+
+$$c_p = \frac{\mathrm{d}q}{\mathrm{d}T} = \frac{\mathrm{d}h}{\mathrm{d}T}$$
+
+
+
+This is because given that pressure, $p$, is constant, work, $w$, can be expressed as:
+
+$$w = -\int^2_1\! p \,\mathrm{d}v = -p(v_2 - v_1)$$
+
+Applying the (1st corollary of the) 1st law to the closed system:
+
+\begin{align*}
+ q + w &= u_2 - u_1 \rightarrow q = u_2 - u_1 + p(v_2 - v_1) \\
+ q &= u_2 + pv_2 - (u_1 + pv_1) \\
+ &= h_2 - h_1 = \mathrm{d}h \\
+ \therefore \mathrm{d}q &= \mathrm{d}h
+\end{align*}
+
+
+
+
+
+
+### Ratio of Specific Heats
+
+$c_p > c_v$ is always true.
+
+
+
+Heating a volume of fluid, $V$, at a constant volume requires specific heat $q_v$ where
+
+$$q_v = u_2 - u_1 \therefore c_v = \frac{q_v}{\Delta T}$$
+
+Heating the same volume of fluid but under constant pressure requires a specific heat $q_p$ where
+
+$$q_p =u_2 - u_1 + p(v_2-v_1) \therefore c_p = \frac{q_p}{\Delta T}$$
+
+Since $p(v_2-v_1) > 0$, $\frac{q_p}{q_v} > 1 \therefore q_p > q_v \therefore c_p > c_v$.
+
+The ratio $\frac{c_p}{c_v} = \gamma$
+
+
+
+# Process and State Diagrams
+
+Reversible processes are represented by solid lines, and irreversible processes by dashed lines.
+
+# Isentropic and Polytropic Processes
+
+## Polytropic Processes
A polytropic process is one which obeys the polytropic law:
@@ -462,7 +462,7 @@ For any values of $x$ and $y$
-# Isentropic
+## Isentropic Processes
*Isentropic* means constant entropy:
@@ -506,7 +506,7 @@ An isentropic process is an idealization of an actual process, and serves as the
real life processes.
They are often desired and often the processes on which device efficiencies are calculated.
-## Heat Transfer During Isentropic Processes
+### Heat Transfer During Isentropic Processes
Assume that the compression 1-2 follows a polytropic process with a polytropic index $n$.
The work transfer is: